Game theory has been studied extensively to answer why cooperation is promoted in human and animal societies. All games are classified into five games: the Prisoner's Dilemma, chicken game (including hawk–dove game), stag hunt game and two trivial games of either all cooperation or all defect, which are studied separately. Here, we propose a new game that covers all five game categories: the weight-lifting game. The player choose either to (1) carry a weight (cooperate: pay a cost) or (2) pretend to carry it (defect: pay no cost). The probability of success in carrying the weight depends on the number of cooperators, and the players either gain the success reward or pay the failure penalty. All five game categories appear in this game depending on the success probabilities for the number of cooperators. We prove that this game is exactly equivalent to the combination of all five games in terms of a pay-off matrix. This game thus provides a unified framework for studying all five types of games. It iEach of these five types of games is known to have a unique structure of dilemmas . BecauseIn this report, we propose a single game called a weight-lifting game that can evaluate all five categories together. In this game, a player has two choices: either carry (lift) a weight (cooperation: C) or pretend to carry it (defection: D); the possible combinations are , ,  and . Both players gain a reward if a weight is successfully carried, but they suffer the penalty of failure if they fail to carry it. The success/failure of carrying the weight depends on the number of cooperators: the probability of success increases with the number of honest lifters.ip . All five types of games are also quantitatively expressed by this game with a certain combination of success probabilities. Thus, the weight-lifting game is mathematically equivalent to all five categories of games. We also discuss the extension of this game to N-person games.This weight-lifting game becomes one of the five types depending on the specific values of the three probabilities 2.c ≥ 0) or defection (D) without any cost. If the baggage is carried successfully, both players obtain a gain (r > 0) irrespective of his/her strategy. For example, the net gain of a player is r − c if he/she cooperates in the successful case. By contrast, the gain is r if he/she is defective but the baggage is successfully carried. In the unsuccessful case, both players pay a fine (f > 0) in addition to the cost c of cooperation. Accordingly, the net gain is − f − c and − f for the cooperator and defector, respectively (a).In the weight-lifting game, baggage is carried by two players randomly selected from among an unlimited well-mixed population. Each player chooses a strategy from two choices: cooperation (C) in carrying the baggage by paying a cost  is the number of cooperators (0 ≤ ip ≤ 1), we introduce two parameters: Δp1 = p1 − p0 (0 ≤ Δp1 ≤ 1) and Δp2 = p2 − p1 (0 ≤ Δp2 ≤ 1). These parameters express increments of the success probability by the presence of one cooperator. The difference between the success probabilities of two cooperators and no cooperator is Δp1 + Δp2 (b). The pay-off matrix of this game is represented in terms of the expected gains for two strategies (c):In terms of the probability of success p1 + Δp2 b. The parategies c:2.1FC=ri = 2 if the two players are both cooperative , i = 1 if one player is cooperative,  and , and i = 0 if neither player is cooperative . The pay-off matrix is simplified by introducing a parameter b = r + f (>0), which is the benefit of a successful game compared to a failed game (d).Note that led game d.a–e black inequality symbols). The first two conditions, R ≥ S and T ≥ P, express that cooperation increases pay-offs. The last condition, T ≥ S, represents that players with different strategies benefit the defector more than the cooperator. The three conditions indicate Δp1 ≥ 0, Δp2 ≥ 0 and c ≥ 0. In terms of the cost-to-benefit ratio c/b, the C-D games are classified according to whether the following inequalities are met .(i)p1 > c/bS > P, i.e. Δ(ii)p2 > c/bR > T, i.e. Δ(iii)p1 + Δp2 > c/b.R > P, i.e. ΔWe consider a general set of C-D games satisfying three conditions R ≥ S, T ≥ P and T ≥ S for the four matrix elements R, S, T and P  is automatically satisfied if either (i) or (ii) is satisfied. Accordingly, among four possibilities for whether or not conditions (i) and (ii) are met, only one (both met) is divided into two cases by the last condition (iii). In total, then, we have the following five cases to consider.Because Δa): R > T ≥ S > P, where all three conditions (i), (ii) and (iii) are met. Both players are cooperative in the Nash equilibrium.Trivial cooperation (TC): a: R > T b): T ≥ R ≥ S > P, where (i) and (iii) are met. A different set of strategies,  and , is the Nash equilibrium.Chicken game (CH): b: T ≥ R c): R > T ≥ P ≥ S, where (ii) and (iii) are met. The two strategies  and  are the Nash equilibrium.Stag hunt (SH): c: R > T d): T ≥ R > P ≥ S, where only (iii) is satisfied. The Nash equilibrium of  is not Pareto optimal.Prisoner's Dilemma (PD): d: T ≥ R e): T ≥ P > R ≥ S, where none of the three conditions (i), (ii) and (iii) is met. Both players choose defection in the Nash equilibrium.Trivial defection (TD): e: T ≥ P f, which indicates that the game type is determined by the three parameters, Δp1, Δp2 and c/b. Every possible C-D game of all five types corresponds uniquely to a certain set of the three parameters .These results are summarized in figure 23.p1, Δp2, b/c) . Instead of c/b, the benefit-to-cost ratio b/c is used as the third axis. While the former is convenient for use in mathematical expressions, the latter has a practical interpretation as an indicator of cooperation. The parameter regions for the five games are shown in colour: TC (yellow), CH (green), SH (red), PD (blue) and TD (violet). As the benefit-to-cost ratio b/c increases, the volume in the parameter space of the cooperative game (TC) increases, while TD and PD, in which cooperation is not promoted, decline. For b/c = 2, TD (violet) and PD (blue) are dominant, while the TC game (yellow) is excluded . For b/c = 5, TC (yellow) becomes dominant, while TD (violet) and PD (blue) shrink . The ratio b/c determines the three boundaries in a Δp1 − Δp2 cross-section . Cooperative behaviour  is promoted when Δp2 exceeds the cost-to-benefit ratio c/b. The relative magnitudes of Δp1 and Δp2 determine the boundary between CH and SH, and the game type becomes SH for Δp2 > Δp1. This means that when the effect of an additional cooperator is increased synergistically, the number of cooperators increases. The CH game is obtained in the opposite case, Δp2 < Δp1.The five types of C-D games are shown in a three-dimensional parameter space  a,b. Instexcluded a,c. For ) shrink b,d. The -section c,d. CoopHere we show the relationship between the game classes  of the current model and the two dilemma strengths introduced recently in the studies of game theory ,22,23. Tp1 = c/b and GID = 0 for Δp2 = c/b. As Δp1 increases, RAD decreases while GID increases. On the other hand, as Δp2 increases, GID decreases and RAD increases. As c/b increases (from figure 4d–f to figure 4a–c), the region in which GID and RAD are both positive enlarges, while the TC region shrinks. Accordingly. Cooperative behaviour becomes difficult to achieve.Here the game class becomes TC when neither of the two dilemmas exists, because there is no dilemma situation. When only GID is positive, the game becomes CH. Conversely, when only RAD is positive, the game becomes SH. If both dilemmas are positive, the game becomes PD game. Note that RAD = 0 for Δ4.p1, Δp2 and the cost-to-benefit ratio (c/b)—the game-theoretical behaviours of three dilemma games  and two trivial games  are obtained in a unique manner. Therefore, players' optimal strategies may change as the parameters Δp1, Δp2 and b/c vary. Because the change in game type represents a change in the strength of the dilemmas [We proposed a weight-lifting game that covers all five types of games. Previously, the resolution of dilemmas has been studied for each individual game . We can now analyse the dilemmas in a unified manner using this single model. Depending on the three parameters—Δdilemmas ,22,23, ip1 and Δp2, which are the increases in the success probability by a single cooperator when there is no or one other cooperator, respectively. The boundary between CH and SH games is determined by their relative magnitude.In the present model, the synergistic effect of cooperation  is repreIt should be noted that CH and SH games address different dilemmas. The gamble-intending dilemma (GID) in the CH game is the dilemma in which players exploit each other. The risk-averting dilemma (RAD) in the SH game is the dilemma of avoiding exploitation by the opponent ,17. It iN-player game appears to be a promising next step because it provides us with two advantages. The public goods game [N players allows us to investigate all possible dilemmas in a unified manner. Another merit is that it allows investigation into how the synergetic effect works depending on the manner in which the success probability changes. A unified treatment of N players and the synergetic effect has not been made before, although the effect of N players and the synergetic effect [Generalization to an ods game  has so fc effect  have beeN-player generalization of the present model, results different from that of the public goods game of a common type may be expected for the synergetic effect [pi increases as the number i of cooperators increases, how the increment in ip varies determines whether synergy is effective or not. If ip increases linearly in i, the public goods game of a common type is obtained, in which cooperation is promoted when the expected utility Δp × (b/c) exceeds 1. By contrast, the success probability ip can be an arbitrary monotonic function in the present model. The synergetic effect depends on how Δip varies. For example, in real-world society, it is expected that pi increases first slowly when there are a small number of cooperators, then moderately as the number increases, and finally slowly again when the cooperators become a dominant majority. In this case, there is a threshold value for the number of cooperators to promote cooperation. Moreover, defection is promoted as the population is saturated with cooperators. It is an interesting future problem to develop these ideas as N-player game simulations of realistic scenarios.In an c effect ,26. WhilThe current model can be applied to the history of social revolution ,28 to di


The Angioedema Quality of Life (AE-QoL) is the first patient reported outcome measure developed for the assessment of quality of life (QoL) impairment in patients with recurrent angioedema (RAE). This study aimed to evaluate the clinimetric properties of the AE-QoL in Thai patients and to establish categories of QoL impairment assessed by the AE-QoL.The validated Thai version of the Dermatology Life Quality Index (DLQI) and Patient Global Assessment of Quality of Life (PGA-QoL) were used to comparatively evaluate the Thai version of AE-QoL. Spearman correlations between the Thai AE-QoL and two other standard measurements (DLQI and PGA-QoL) were investigated to determine convergent validity. The Thai DLQI and PGA-QoL were used to categorize patients according to their QoL. Known-group validity of the Thai AE-QoL was later analyzed. The reliability of the Thai AE-QoL was investigated using Cronbach’s alpha and intraclass correlation. Three different approaches including the distribution method, receiver operating characteristic curve analysis, and the anchor based-method were used for the interpretability.A total of 86 patients with RAE with a median age of 38.0 ± 15.1 years (range 18–76) were enrolled. Of those, 76 patients (88%) had RAE with concomitant wheals, and 10 patients (11.6%) had RAE only. The AE-QoL assessed RAE-mediated QoL impairment with high convergent validity and known-groups validity, high internal consistency and test-retest reliability, and good sensitivity to change. Although the AE-QoL did not differentiate between patients with moderate and large effect as measured by PGA-QoL or DLQI in this study, AE-QoL total values of 0–23, 24 to 38, and ≥ 39 could define patients with “no effect”, “small effect”, and “moderate to large effect” of RAE on their QoL, respectively.This study supports the validity and reliability of the Thai version of the AE-QoL, which is a very different language from the original version. Categories allow to classify the effect of RAE on patients’ QoL as “none”, “small”, and “moderate to large”. Further studies are needed to confirm the applicability of AE-QoL in other Asian populations”. Recurrent angioedema (RAE) is characterized by the unpredictable, sudden and recurrent onset of nonpitting swelling of the skin, subcutaneous tissues, and/or mucous membranes . AngioedRAE is classified into mast cell mediator-mediated and bradykinin-mediated, based on the underlying pathogenic mechanisms . In clinFor assessing HRQoL of angioedema patients, the Angioedema Quality of Life Questionnaire (AE-QoL)  and the As of now, there is the lack of and need for categorization/banding of HRQoL impairment in RAE patients. Also, we are lacking validated version of patient-reported outcome (PRO) outcome measures for assessing QoL impairment in Asian patients with RAE. Accordingly, this study generated the Thai version of the AE-QoL, evaluated its clinimetric properties in an Asian population of RAE patients, specifically in Thailand, and established categories of AE-QoL-assessed QoL impairment in this RAE population.Formal permission was given by MOXIE GmbH  to translate the original German version of the AE-QoL into Thai, and to validate the translated instrument . The AE-Misconceptions or misinterpretations during the translation process were resolved via a cooperative review and editing process that included the Thai research team and the instrument developers. After consensus was reached among all assessors, cognitive debriefing was performed with representatives of the target population and target language group to evaluate if the responders understand the questionnaire as intended , 18. TheThis study was approved by the Siriraj Institutional Review Board (SIRB), Faculty of Medicine Siriraj Hospital, Mahidol University, Bangkok, Thailand (COA no. Si271/2017). Adult Thai RAE patients aged 18 years or older attending the Urticaria Center of Reference and Excellence  of the DThe DLQI is a general dermatologic HRQoL questionnaire that was established by Finlay et al. in 1994 . The DLQThe PGA-QoL is a patient evaluation instrument for assessing HRQoL during the preceding four-week period. A 5-point Likert-scale is used to rate level of effect, as follows: 0 = no effect, 1 = mild effect, 2 = moderate effect, 3 = severe effect, and 4 = very severe effect , 23.At baseline, patients were informed how to complete the following questionnaires: Thai AE-QoL, Thai DLQI, and PGA-QoL. After patients acknowledged understanding of how to complete all three questionnaires, they were asked to complete the Thai AE-QoL, Thai DLQI, and PGA-QoL by themselves at the clinic. After 4 weeks of treatment (2nd visit), all of the questionnaires were once again completed by the patients in the clinic.P- values of less than or equal to 0.05 were considered to be statistically significant.Consensus-based Standards for the Selection of Health Measurement Instruments (COSMIN), a standard protocol for evaluating the methodologic quality of studies in health measurement instruments, was applied in this study . The StaConvergent validity indicates the correlation between the Thai AE-QoL and other standard measurements for HRQoL (DLQI and PGA-QoL). Spearman correlation coefficients of < 0.3, 0.3–0.6, and > 0.6 were considered weak, moderate, and strong correlations, respectively .Known-group validity demonstrates the ability of the AE-QoL to discriminate groups that are assumed to be different. For HRQoL, patients were classified into three groups using the DLQI and PGA-QoL scores in this study, as follows: (i) ‘no effect’ ; (ii) ‘small effect’ ; and, (iii) ‘moderate to large effect’ .Cronbach’s α reliability coefficient was used to analyze internal consistency. Excellent, good, and acceptable reliability were defined as α values of ≥0.9, 0.7 ≤ α < 0.9, 0.6 ≤ α < 0.7, respectively . InternaTest-retest reliability assesses the consistency of scores across multiple administrations. Stable patients should have comparable scores at two different administrations. Data from patients with no change in PGA-QoL score (stable in HRQoL) during 4 weeks of treatment was used to determine test-retest reliability. Intraclass correlation coefficient (ICC) values of < 0.40, 0.4–0.75, and > 0.75 indicate poor, average, and strong reliability, respectively .Sensitivity to change is defined as the ability of an instrument to detect change over time, regardless of whether the change is meaningful . Correlai.Distribution method (MCID-1) indicates the numerical distribution of values. The standard error of measurement (SEM) is widely accepted to represent the MCID of an instrument –34. The ii.Receiver operating characteristic (ROC) curve analysis (MCID-2) shows the sensitivity and specificity over the range of absolute reductions in scores that can be used to identify “improvement” , 35. Areiii.Anchor-based approach (MCID-3) compares changes in scores to an anchor value. MCID-3 is the difference between the mean “improvement” score and the mean “worsening/stable” score. Patients with a decrease in PGA-QoL score of ≥1 and patients with an increased or unchanged PGA-QoL score were defined as ‘improvement’ and ‘worsening or stable’ HRQoL, respectively.Interpretability is the ability of an instrument to transform qualitative meaning to quantitative scores, i.e. the degree to which the values obtained by the use of the instrument produce information relevant to the patient and the physician in relation to the measured construct. Interpretability is composed of the minimal clinically important difference (MCID) and categorization. The MCID is defined as the smallest difference in a score that a patient can recognize as meaningful improvement , 31. DifCategorization is the process of determining cutoff scores for a measurement instrument that are then used to categorize patients into different groups. The DLQI and PGA-QoL were used to categorize patients into “no effect” , “small effect” , “moderate effect” , and “large and very large effect” .A total of 86 patients with RAE with a median age of 38.0 ± 15.1 years (range 18–76) were enrolled in this study. Of those, 76 patients (88%) had RAE with concomitant wheals, and 10 patients (11.6%) had RAE only. Of 10 patients who had RAE without wheals, 9 and 1 patients had idiopathic angioedema, and angiotensin-converting enzyme inhibitor-induced angioedema, respectively. Demographic data, scores at baseline, and treatments are shown in Table p < 0.0001) and between AE-QoL total score values and PGA-QoL values .The AE-QoL showed high convergent validity, with strong positive correlations between AE-QoL and DLQI total score values  and with a small effect and a moderate to large effect  were significant. Overall, there were also significant differences in the mean AE-QoL score among patients with no effect, small effect, and moderate to large effect on HRQoL (p < 0.0001). However, the AE-QoL did not differentiate between patients with moderate and large effect as measured by PGA-QoL or DLQI. A statistical significance was not achieved when using the DLQI or the PGA-QoL to classify patients as those with “no effect”, a “small effect”, a “moderate effect”, and a “large effect”.The AE-QoL also showed high known-groups validity as it was able to discriminate among patients who showed differences in HRQoL impairment as assessed by the use of the PGA-QoL and the DLQI. Figure All Cronbach’s α values were higher than 0.72, which indicates good to excellent internal consistency of the Thai AE-QoL. Forty-four patients who had no change in HRQoL (no change in PGA-QoL score) during 4 weeks were analyzed for intraclass correlation coefficients (ICCs). The ICC was 0.76 for the AE-QoL total score, which indicates strong intra-rater reliability. The ICCs (range: 0.64–0.75) for the AE-QoL domains were average (Table p < 0.0001). This allowed us to further analyze MCID.The correlation between AE-QoL score changes and changes in PGA-QoL and DLQI were 0.39 and 0.42, respectively  and specificity (76.7%) for defining patients with meaningful improvement in HRQoL  of the AE-QoL, a change in the total score of > 6 was identified as the MCID . It is gAccording to the results of previous study, the linear correlation between AE-QoL total score and self-rated HRQoL impairments was well-demonstrated . HoweverThe maximum values to classify patients into “no effect” group were 23 and 25 using PGA-QoL and DLQI, respectively. The maximum value of 23 should be used as the cutoff score for the “no effect” category, as it had a higher AUC value than that of 25. Similarly, the minimum values to classify patients as those with a moderate to large effect were 38 and 36 using PGA-QoL and DLQI, respectively. The value of 38 should be chosen the cutoff value, as it had a higher AUC than that of 36. Thus, we propose that AE-QoL total score values from 0 to 23, from 24 to 38, and ≥ 39 define patients with “no effect”, a “small effect”, and a “moderate to large effect” of their RAE on their HRQoL, respectively. To provide a clinical perspective on this AE-QoL categorization: In the recent X-ACT study, in CSU patients with angioedema refractory to H1-antihistamine, patients had a “very large” negative impact on their HRQoL as reflected by the mean DLQI total score of 14.6 at baseline. The mean AE-QoL total score at baseline of these patients was 56.2. Using our AE-QoL categorization, the patients in this study fall into the category “moderate to large effect” of RAE on HRQoL. After 12 weeks of omalizumab treatment in this study, the mean AE-QoL total score declined significantly from 56.2 to 27.7, which changed the AE-QoL category of patients from “moderate to large effect” to “small effect” of RAE on HRQoL .The strengths of this study include the use of three different approaches to establish the MCID of the Thai AE-QoL. However, several limitations should be noted including its small study population and the very high predominance of RAE patients with CSU. Secondly, no patients with HAE were enrolled in this study. Angioedema activity in HAE is generally more severe than in other causes of angioedema. Kulthanan, et al. reported that HAE was observed in 1 out of 105. Lastly, the AE-QoL did not differentiate between patients with moderate and large effect as measured by PGA-QoL or DLQI in this study. Further studies in larger study populations that include patients with high disease activity and causes of RAE other than CSU are needed to confirm the categorization of the AE-QoL.”The Thai version of AE-QoL was found to be a valid and reliable instrument for assessing disease–specific impact on QoL in Thai RAE patients. A change in AE-QoL of > 6 points defines a meaningful change in HRQoL. RAE patients should be categorized for the impact of their disease on their QoL as those with “no effect”, “small effect”, and “moderate to large effect” based on AE-QoL total score values of up to 23, 24–38, and 39 and higher, respectively.


Nature Communications 10.1038/s41467-018-07150-9; published online 07 November 2018Correction to: In the original version of this Article, the label “RTK” in Fig. 6a was inadvertently changed to “RTE”. This has now been corrected in the PDF and HTML versions of the Article.


Deep-learning based methods have succeed in modeling in-vivo protein-DNA binding, but they often (1) follow the fully supervised learning framework and overlook the weakly supervised information of genomic sequences that a bound DNA sequence may has multiple TFBS(s), and, (2) use one-hot encoding to encode DNA sequences and ignore the dependencies among nucleotides. In this paper, we propose a weakly supervised framework, which combines multiple-instance learning with a hybrid deep neural network and uses k-mer encoding to transform DNA sequences, for modeling in-vivo protein-DNA binding. Firstly, this framework segments sequences into multiple overlapping instances using a sliding window, and then encodes all instances into image-like inputs of high-order dependencies using k-mer encoding. Secondly, it separately computes a score for all instances in the same bag using a hybrid deep neural network that integrates convolutional and recurrent neural networks. Finally, it integrates the predicted values of all instances as the final prediction of this bag using the Noisy-and method. The experimental results on in-vivo datasets demonstrate the superior performance of the proposed framework. In addition, we also explore the performance of the proposed framework when using k-mer encoding, and demonstrate the performance of the Noisy-and method by comparing it with other fusion methods, and find that adding recurrent layers can improve the performance of the proposed framework.Modeling  Modeling 3, dramatically increases the amount and spatial resolution of available data, which is helpful for the in-depth study of in-vivo protein-DNA binding. However, DNA sequences directly extracted from ChIP-seq cannot precisely represent TFBS since the outputs of such experiments contain a lot of noisy4. Thus lots of methods have been developed for precisely predicting protein-DNA binding sites, including conventional algorithms9 and deep-learning based methods12. Not surprisingly, deep-learning based methods are better than conventional algorithms at modeling protein-DNA binding. DeepBind10 and DeepSea11 were two famous deep-learning based methods, which used convolutional neural network (CNN) to model the binding preference of DNA-proteins with a superior performance over conventional methods. DanQ12 designed a hybrid deep neural network to quantify the function of DNA sequences, which first used a convolutional layer to detect regulatory motif features from DNA sequences, and subsequently employed a bi-directional recurrent layer to capture long-term dependencies between motif features. Soon after, a number of deep-learning based methods are proposed for modeling in-vivo protein-DNA binding17.In the past decades, the introduction of high-throughput sequencing technologies, especially ChIP-seq.in-vivo protein-DNA binding, they usually overlook the weakly supervised information of genomic sequences that a bound DNA sequence may have multiple TFBS(s). In consideration of this information, Gao et al.18 developed a multiple-instance learning (MIL) based algorithm, which combines MIL with TeamD19, for modeling protein-DNA binding, and recently Zhang et al.20 also developed a weakly supervised convolutional neural network (WSCNN), which combines MIL with CNN, for modeling protein-DNA binding. Moreover, they are inclined to use one-hot encoding to encode DNA sequences, which means that it only considers the independent relationship among nucleotides. However, recent studies have shown that taking into consideration the high-order dependencies among nucleotides can improve the performance of modeling protein-DNA binding23. In consideration of this information, Zhou et al.24 evaluated DNA-binding specificities based on mononucleotide (1-mer), dinucleotide (2-mer), and trinucleotide (3-mer) identity, and stated that 2-mer and 3-mer may contain implicit DNA shape information and partially capture the effect of the DNA shape variation on binding. Zhang et al.25 proposed a high-order convolutional neural network, which first used k-mer encoding to transform DNA sequences into image-like inputs of high-order dependencies, and then applied CNN to extract motif features from these inputs.Although deep-learning based methods have achieved remarkable performance on modeling Noisy-and26 to replace them. (3) WSCNN, like other deep-learning based methods, used one-hot encoding to transform DNA sequences into image-like inputs. However, the relationship between nucleotides is not independent in practice. Therefore we use k-mer encoding to transform DNA sequences into image-like inputs of high-order dependencies, and explore the performance of the proposed framework when using dinucleotide (2-mer) and trinucleotide (3-mer) as inputs in the weakly supervised framework. In summary, the proposed framework firstly use the concepts of MIL to segment DNA sequences into multiple instances, and adopt k-mer encoding to transform sequences into image-like inputs of high-order dependencies, and then design a hybrid neural network to compute a score for all instances, and finally employ the Noisy-and method to fuse the predicted values of all instances as the final prediction of a bag. We conducted a lot of comparative experiments on in-vivo datasets to show that our proposed framework outperforms other competing methods. Besides, we also show the performance gain of the proposed framework when using k-mer encoding, and compare the performance of the Noisy-and method with other fusion methods, and demonstrate the effectivness of adding recurrent layers.Inspired by the above observation, we extend our previous work WSCNN from three aspects in this paper. (1) WSCNN mainly employed CNN to learn motif features from DNA sequences, and did not take into consideration the long-term dependencies between motif features. In the weakly supervised framework, therefore we add a bi-directional recurrent layer after the convolutional layer to capture the forward and backward long-term dependencies between motif features. (2) WSCNN attempted to use four fusion methods to fuse the predicted values of all instances in a bag, and then selected the best one of them as the final prediction. However, it is inconvenient for user to decide which one is better, so they have to try the four fusion methods one by one. Therefore we offer a better and more robust fusion method Noisy-and in Section II. We give a detailed analysis of the experimental results, and discuss the hyper-parameter settings in Section III.The rest of the paper is organized as follows. We give a detailed description of the proposed framework, and introduce the fusion method in-vivo protein-DNA binding. Actually, the task can be thought of as a binary classification problem that separates positive sequences (bound) from negative sequences (non-bound). The output of the network is a probability  distribution over two labels (1/0), since a binary classification problem can be addressed also through a binary output (1/0). This framework includes three stages in general: data processing, model designing, and results merging.In this section, we give a detailed description of the proposed framework for modeling 20, which ensures that (1) the weakly supervised information can be retained, and that (2) a large amount of instances containing TFBS are generated. This method is defined as a sliding window of length c, which divides DNA sequences of length l into multiple overlapping instances by a stride s. A bag is composed of all possible instances in the same sequence, and the number of instances in this bag is s and c are two hyper-parameters that need to be tuned by cross-validation. If (l − c) is not a multiple of s, we pad ‘0’ at the end of DNA sequences.Considering the weakly supervised information of DNA sequences, thus it is reasonable to use the concepts of MIL to deal with DNA sequences. Therefore we divided them into multiple overlapping instances following the worksk-mer encoding method25 to transform all instances into image-like matrices of high-order dependencies. This method can be implemented according to ; 2-mer encoding: taking into consideration the dependencies between two adjacent nucleotides, and each dinucleotide is mapped into a vector of size 16 ; 3-mer encoding: taking into account the dependencies among three adjacent nucleotides, and each trinucleotide is mapped into a vector of size 64 .After segmenting DNA sequences, all instances should be transformed into image-like inputs that can be handled by CNN. One-hot encoding is a commonly-used method in deep-learning based methods, but it ignores high-order dependencies among nucleotides. In order to capture the dependencies, therefore we use the rding to :1\documek = 1 is shown in Fig. l = 10, c = 8, s = 1, and the red dashed box denotes a sliding window of length c = 8. Through this stage, DNA sequences can be encoded into image-like inputs that can be easily handled by CNN.A graphical illustration of data processing when k × 1 × (c − k + 1) (batchsize × channel × height × width), and then all instances of a bag can be concatenated along the height axis. Therefore a bag can be represented by a tensor of shape 1 × 4k × n × (c − k + 1), where n is the number of instances per bag  is another variant of ANN where connections between neurons form a directed graph. Unlike CNN, RNN can use its internal state (memory) to exhibit dynamic temporal or spatial behavior. In the designed model, the convolution layer is used to capture motif features, while the recurrent layer is used to capture long-term dependencies between the motif features. The model is arranged in this order: a convolutional layer → a max-pooling layer → a dropout layer → a bi-directional recurrent layer → a dropout layer → a softmax layer.Considering the spatial and sequential characteristics of DNA sequences, we design a hybrid deep neural network, which integrates convolutional and recurrent neural networks in this stage. Convolutional neural network (CNN) is a special version of article neural network (ANN)30 (ReLU) layer. The early work13 has explored the performance of using different number of convolutional kernels, and found that adding more kernels can significantly improve performance. Thus the number of kernels was set to a fixed value 16 in the proposed framework.This layer is used to capture motif features, which can be thought of as a motif scanner to compute a score for all potential motifs, and often followed by a rectified linear unitBoth DeepBind and DeepSea used a global max-pooling layer to pick out the maximum response of the whole sequence, while our deep model uses a max-pooling layer of a certain size  to keep the local best values of the whole sequence.31 is a widely-used regularization technique for reducing overfitting in deep neural networks by preventing complex co-adaptations on data, which randomly sets the outputs of the previous layer to zero with a dropout ratio. The dropout ratio is a hyper-parameter that was investigated by cross-validation in the experiments.Dropout strategy32 is used. A LSTM unit usually consists of a cell, an input gate, a forget gate and an output gate, where the cell remembers values over arbitrary time intervals and the three gates regulate information flows into and out of the cell. In this paper, we did not use a fully-connected layer to follow this layer, as this will result into worse performance. The number of neurons in this layer was set to 32, so the output size of this layer is 64.In order to capture the forward and backward long-term dependencies between the motif features, a bi-directional recurrent layer composed of long short-term memory (LSTM) unitsIn order to get a probability distribution over two labels which separately represent bound or non-bound sequences, a softmax layer is used in this model. It is composed of two neurons, each of which is densely connected with the previous layer and computes a probability.Max function is frequently used as the fusion function in MIL. But Max only focuses on the most informative instance and overlooks other instances that may contain useful information. Therefore WSCNN used three additional fusion methods  to utilize all instances that may contain useful information. However, both Average and Linear Regression take advantage of all information, inevitably containing useless information, and Top-Bottom Instances needs to manually determine the number of the highest and lowest scoring instances. Moreover, how to effectively take advantage of abundant positive instances is also a key point. To solve the above problems, we find a better and more elegant fusion method, named Noisy-and26, which is based on a different assumption that a bag is labeled as positive if the number of positive instances in the bag exceeds a threshold. This method is defined as follows:Pi,j denotes the score of the j-th instance at the i-th bag, and ni denotes the number of instances in the i-th bag, and n instances in the i-th bag. Noisy-and is designed to activate a bag level probability Pi once the mean of the instance level probabilities a is a fixed hyper-parameter that controls the slope of Noisy-and. bi represents an adaptable soft threshold for each class i and needs to be learned during training. σ(a(1 − bi)) and σ(−abi) are included to normalized Pi to  for bi in  and a > 0.MIL is commonly based on an assumption that a bag is labeled as positive if there is at least one instance that contains TFBS, and is labeled as negative if there are no any instances that contain TFBS. Therefore the Through this stage, the predicted values of all instances in a bag are fused to yield a final prediction (probability) over ‘bound’ and ‘non-bound’ labels.k-mer encoding) → a convolutional layer → a max-pooling layer → a dropout layer → a bi-directional recurrent layer → a dropout layer → a softmax layer → a fusion layer. A graphical illustration of the proposed framework is shown in Fig. In summary, the proposed framework is arranged in this order: data processing . For each public dataset, an average number of ~15000 top ranking sequences were chosen as the positive data where each sequence is composed of 200 bps, and the corresponding negative data was generated by matching the repeat fraction, length and GC content of the positive ones following the work10, DanQ12, and WSCNN20, respectively.To better evaluate the performance of WSCNNLSTM, we constructed three deep-learning based models, which are similar to DeepBindModel 1: This model is a single-instance learning (SIL) based method, and has the similar architecture to DeepBind. It is arranged in this order: data processing (one-hot encoding) → a convolutional layer → a global max-pooling layer → a fully connected layer → a dropout layer → a softmax layer.Model 2: This model is a single-instance learning (SIL) based method, and has the similar architecture to DanQ. It is arranged in this order: data processing (one-hot encoding) → a convolutional layer → a max-pooling layer → a dropout layer → a bi-directional recurrent layer → a dropout layer → a softmax layer.Model 3: This model is a multiple-instance learning (MIL) based method, and has the similar architecture to WSCNN. It is arranged in this order: data processing (segmentation + one-hot encoding) → a convolutional layer → a global max-pooling layer → a fully connected layer → a dropout layer → a softmax layer → a fusion layer.46.To comprehensively assess the performance of WSCNNLSTM, we adopted three standard evaluation metrics in this paper, including area under receiver operating characteristic curve (ROC AUC), area under precision-recall curve (PR AUC), and F1-score, which are widely used in machine learning and motif discovery47 and PR AUC two commonly-used metrics in which PR AUC is often used under the situation of imbalanced data. Since PR AUC needs not to consider the number of true negative samples, thus it is less prone to influenced by the class imbalance than the ROC AUC metric is48.ROC AUC49.F1-score is a solid metric to measure the classification performance of the classifier, which simultaneously takes into consideration the precision and the recall when computing the score of a testhttps://github.com/turningpoint1988/WSCNNLSTM. The parameters in the deep-learning based methods were initialized by Glorot uniform initializer50, and optimized by AdaDelta algorithm51 with a mini-batchsize of 300. For some sensitive hyper-parameters , we selected the best configuration using a grid-search strategy. Epochs of training were set to 60, where after each epoch of training, the accuracy of the validation set was assessed and monitored, and the model with the best accuracy in the validation set was saved. The instance length c and segmentation stride s were set to 120 and 10, since WSCNN has stated that the two hyper-parameters has little effect on the main conclusion. The hyper-parameter a in Noisy-and was set to 7.5 following the work26. The hyper-parameter settings of all the deep-learning based methods in this paper are detailed in Table We implemented WSCNNLSTM and the competing methods by Keras with the tensorflow backend, which are freely available at: in-vivo ChIP-seq datasets is shown in Fig. A comparison of WSCNNLSTM and the competing methods on 50 Figure k-mer encoding method can take into consideration the high-order dependencies among nucleotides, which may improve the performance of modeling in-vivo protein-DNA binding. As stated by Zhou et al.24, 2-mer (dinucleotide) and 3-mer (trinucleotide) may contain implicit DNA shape information and partially capture the effect of the DNA shape variation on binding. Moreover, Zhang et al.25 has stated that the number of learnable parameters will grow exponentially with the increase of k. Therefore, in order to make a trade-off between performance and computational complexity, we transformed DNA sequences into matrices that consist of 2-mer or 3-mer by setting k to 2 or 3 in k-mer encoding.Unlike one-hot encoding, the k-mer encoding in the weakly supervised framework, we carried out some comparative experiments on 23 ChIP-seq datasets from the Gm12878 cell line, and the detailed results are shown in Supplementary Tables To test the performance of k-mer encoding in WSCNN under the ROC AUC, PR AUC, and F1-score metrics. We can find that the performance of 2-mer and 3-mer encoding is much better than that of one-hot encoding, demonstrating the effectiveness of k-mer encoding for modeling in-vivo protein-DNA binding. Figure k-mer encoding in WSCNNLSTM under the ROC AUC, PR AUC, and F1-score metrics. We can also find the same trend that the performance of 2-mer and 3-mer encoding is much better. Figure k-mer encoding is superior to one-hot encoding. Moreover, we find that the performance of WSCNN and WSCNNLSTM is improved with the increase of k. We think that the reason of the good performance may lie in: it explicitly considers the high-order dependencies among nucleotides (which contains implicit DNA shape information).Figure Max, Linear Regression, Average and Top-Bottom Instances) to fuse the predicted values of all instances in a bag, and then selected the best one as the final prediction. However, Max only focuses on the most informative instance and overlooks other instances that may contain useful information, and both Average and Linear Regression take advantage of all information, inevitably containing useless information, and Top-Bottom Instances needs to manually determine the number of the highest and lowest scoring instances. Moreover, how to effectively take advantage of abundant positive instances is also a key point. Thus we adopt a better and more elegant fusion method, named Noisy-and, in this paper.WSCNN employed four fusion methods (Noisy-and in the weakly supervised framework, we carried out some comparative experiments on 23 ChIP-seq datasets from the Gm12878 cell line, and the detailed results are shown in Supplementary Tables To test the performance of Noisy-and and Max, Average in WSCNN under the ROC AUC, PR AUC, and F1-score metrics. We can find that the performance of Noisy-and is much better than that of Max and Average. Figure Noisy-and and Max, Average in WSCNNLSTM under the ROC AUC, PR AUC, and F1-score metrics. We find that the performance of Noisy-and is also much better than that of Max and Average. Figure Noisy-and that a bag is labeled as positive if the number of positive instances in the bag exceeds a threshold.Figure in-vivo protein-DNA binding, but they ignores the long-term dependencies between motifs. Therefore we add a bi-directional recurrent layer after the convolutional layer in the weakly supervised framework, just like DanQ does, to capture the forward and backward long-term dependencies between motifs.Convolutional layers are often used as motif scanners to capture motifs in the task of modeling Figure in-vivo protein-DNA binding. The proposed framework contains three stages: data processing, model designing, and results merging, where the first stage contains the segmentation process and k-mer encoding, and the second stage contains a hybrid deep neural network, and the last stage contains the Noisy-and fusion method. The experimental results on in-vivo ChIP-seq datasets show that our proposed framework performs better than the competing methods. In addition, we also explore the performance of the proposed framework when using k-mer encoding and show that the k-mer encoding can significantly improve the performance of modeling in-vivo protein-DNA binding, and demonstrate that the Noisy-and function is a better fusion method in the weakly supervised framework.In this paper, we propose a weakly supervised framework, which combines multiple-instance-learning with a hybrid deep neural network, for modeling k. However, the big k will bring out exponentially growing learnable parameters, and the performance of models may be degenerated when k reaches a certain value. Therefore, we will explore the performance of k-mer encoding with a big k value, and propose a corresponding solution to it in the future works.From the above results, we find that the performance of WSCNN and WSCNNLSTM is improved with the increase of Supplementary Materials


Pan paniscus) and chimpanzees (Pan troglodytes). As in most mammals, female bonobos and chimpanzees have longer life expectancy than males.Throughout life, physiological homeostasis is challenged and the capacity to cope with such challenges declines with increasing age. In many species, sex differences exist in life expectancy. Sex-specific differences have been related to extrinsic factors like mate competition and/or intrinsic proximate mechanisms such as hormonal changes. In humans, an intrinsic factor related to aging is soluble alpha klotho (α-Kl). Both sexes show an age-related decline in α-Kl, but throughout life women have higher levels than men of the same age. Sex differences in α-Kl have been linked to a shorter lifespan, as well as to specific morbidity factors such as atherosclerosis and arteries calcifications. In non-human animals, information on α-Kl levels is rare and restricted to experimental work. Our cross-sectional study is the first on α-Kl levels in two long-lived species: bonobos  contains supplementary material, which is available to authorized users. Some have suggested that ageing is too complicated for serious scientific study, or that it is like a slow-motion car crash — everything just gets wrecked” [“wrecked” .Aging is defined as the age-related deterioration in physiological functions necessary for survival and fecundity –4. IndivPan paniscus) and chimpanzees (Pan troglodytes) are sister species and share life history traits such as slow maturation, late onset of reproduction [Bonobos  [When trying to explain sex-specific aging patterns, intrinsic markers are of strong interest. Investigating physiological markers that change with age is promising to help explain the sex-specific aging patterns observed in the two IGFBP-3) . This prIGFBP-3) –35. In bIGFBP-3) . FurtherMacaca mulatta) [One marker for detecting age-related changes in physiology in humans is soluble alpha klotho (α-Kl). The Klotho gene was originally described in a mutant mouse strain that could not express klotho . The klomulatta) .The human Klotho gene encodes either a transmembrane protein or a secreted form . The secIn animals, observed sex differences in life expectancy have also been linked to extrinsic factors. For instance, the intensity of male-male competition has a strong impact on sex differences in life expectancy, with males in polygynous societies experiencing a reduced lifespan compared to monogamous species , 60. MalThe aims of this study were to assess (a) if bonobos and chimpanzees experience an age-related decline in α-Kl levels, and (b) if α-Kl levels are associated with sex differences in life expectancy. We hypothesized that α-Kl levels would be lower in older individuals in comparison with younger ones, and lower in males compared with females. In case α-Kl levels reflect trajectories of aging that are due to extrinsic factors such as intensity of intra-sexual competition , we hypoP < 0.001). However, a likelihood ratio test revealed that the three-way interaction of sex, species, and age was not significant . We ran a reduced model, excluding the three-way interaction, while including all two-way interactions  and the random effects. In this reduced model none of the two-way interactions were significant . Further reducing the model revealed that the interactions of sex and age at sampling as well as species with age at sampling dropped out non-significant, while age at sampling remained as a main effect. This final reduced all-age-model revealed that with increasing age, α-Kl levels decreased significantly (Table We ran a linear mixed model (LMM ) to examN = 140, rho = − 0.632, p <  0.001).We correlated age at sampling time with α-Kl levels in chimpanzees and bonobos. A Spearman correlation of rank showed a significant negative correlation of α-Kl levels with age (P <  0.001), indicating that the test predictors age, sex, and species significantly influenced α-Kl levels in bonobos and chimpanzees older than 15 years. In this adults-only-model, the likelihood ratio test revealed that the three-way interaction of age, sex, and species was not significant . We ran a reduced adults-only-model with all two-way interactions  and random effects. As in the all-age-model, no two-way interaction including age at sampling was significant . However, the interaction of species and sex showed a significant effect on α-Kl levels , indicating that α-Kl levels differ between the two species and in a sex-specific way. Therefore, interactions with age at sampling day were removed, but the interaction of sex*species as well as age at sampling day as a main effect were kept in the second reduced adults-only-model. The results of the final reduced adults-only-model (including age at sampling and the sex*species interaction) revealed that also in adult individuals, α-Kl levels decline significantly with increasing age, and that males and females show significant differences in α-Kl patterns within a species (Table In a second LMM we investigated sex and species differences in α-Kl levels during adulthood (adults-only-model). In the adults-only-model, the full model was significantly different from the null model   . One possible explanation is that our sample pool included individuals from all ages whereas measures obtained for human studies are restricted to specific age categories, namely either children [As in humans, α-Kl levels of bonobos and chimpanzees decreased significantly with age , 75–77,  − 0.255 ) in humachildren , adults children , or oldechildren . AnotherBased on data from human studies , 76, 78,Pan species are explained by sex-specific exposure to stressors in relation to their social system? Prather et al. [One compelling question that our results raise is whether the differences in α-Kl levels between the sexes of the two r et al.  exploredr et al. ; cortisor et al. , 80, it r et al.  had alsor et al. . If the r et al. –83. Furtr et al. , 85. In r et al. , 87. Ther et al. , and aggr et al. . In addir et al. . Howeverr et al. , howeverr et al. –96. To wIn bonobos and chimpanzees, α-Kl levels decline with age suggesting that potential may arise from this marker for studying the process of aging on an individual, population, species, and on a comparative level. Bonobo females have higher α-Kl levels in comparison to males, which correspond to sex differences in life expectancy and mirrors sex differences found in humans. In chimpanzees, males have higher α-Kl levels than females posing a mismatch with sex differences in life expectancy. The observed sex differences α-Kl levels correspond with inter-sexual dominance relations of the two species and we cannot exclude that social stress and related cortisol levels affect α-Kl levels. Thus, considering interactions between α-Kl and other hormones, like glucocorticoids and AVP, potentially helps to explain species differences in α-Kl levels and how this relates to life expectancy and physiological aging.ad libitum access to fresh water, and food was offered at least three times a day and consisted mainly of a mixture of fruits and vegetables. All subjects had access to indoor and outdoor enclosures.We measured soluble α-Kl levels in 140 serum samples of 63 bonobos  and 77 chimpanzees  from 16 zoos  and 3 to 49 years for chimpanzees (average age 22.9 years). For 54 (86%) of the 63 bonobos, the exact birthdates were known; exact birthdates were available for 63 (82%) of the chimpanzees. For one male and two female chimpanzees for whom only the year and month of birth were known, we set the day of birth to the 15th of the respective month. For nine bonobos  and eleven chimpanzees , exact age was not known and the day of birth was set optionally to June 15th of the estimated year of birth. Individuals for whom the age was estimated were mainly older and therefore the relative error of estimated age was relatively small.For the analyses of α-Kl levels during adulthood, we created a subset with samples of individuals older than 15 years of age , 98. ThiN = 3), and the intra-assay coefficient of variation was 3.2%. Based on these results, the ELISA was considered to be appropriate for measuring α-Kl in serum samples of bonobos and chimpanzees. Two and more freezing cycles changed measured α-Kl levels by around 10% . The assay was originally developed for the measurement of soluble α-Kl in human serum and EDTA (Ethylenediaminetetraacetic acid)-plasma samples. Before we used the ELISA for routine measurements, we tested for parallelism of diluted pool serum samples for each species separately. Pooled samples consisted of one female and one male sample of each species. Serially diluted pooled samples were found to parallel the standard curve , one toIn the all-age-model investigating α-Kl levels during lifetime, the response variable, α-Kl, was log-transformed to approximate a normal distribution. The model included a three-way interaction of age (z-transformed), sex, and species (both dummy coded), as well as relevant random effects (see below). The three-way interaction was included in the all-age full model as mortality and life expectancy was expected to differ between sexes and species, and α-Kl is associated with human aging syndromes. We predicted that α-Kl levels decline with age in both species but at different times, or, depending on sex, with different steepness, leading to species-specific patterns that vary between the sexes. We included random intercepts for zoo-ID, to account for a possible influence of relevant animal husbandry conditions, and for subject, as eight individuals were included twice in the data set. There was considerable variation in terms of demography of ape groups across different zoo facilities; therefore, we included random slopes for age in zoo-ID.Model assumptions  were assessed by visual inspections of a histogram, a q–q plot of the residuals, and by plotting residuals against fitted values. All model assumptions were met. We examined Variance Inflation Factors (VIF ) using tP = 0.05.To investigate the significance of the predictors age, species, and sex and their interactions, we compared the full model with a null model including only the random effects using a likelihood ratio test


Minimum Vertex Cover andMaximum Clique. Finally, we use the conceptof well-structured modulators to develop an algorithmic meta-theorem for decidingproblems expressible in monadic second order logic, and prove that this result istight in the sense that it cannot be generalized to LinEMSO problems.A modulator in a graph is a vertex set whose deletion places theconsidered graph into some specified graph class. The cardinality of a modulator tovarious graph classes has long been used as a structural parameter which can beexploited to obtain fixed-parameter algorithms for a range of hard problems. Here weinvestigate what happens when a graph contains a modulator which is large but“well-structured” (in the sense of having bounded rank-width). Can such modulatorsstill be exploited to obtain efficient algorithms? And is it even possible to findsuch modulators efficiently? We first show that the parameters derived from suchwell-structured modulators are more powerful for fixed-parameter algorithms than thecardinality of modulators and rank-width itself. Then, we develop a fixed-parameteralgorithm for finding such well-structured modulators to every graph class which canbe characterized by a finite set of forbidden induced subgraphs. We proceed byshowing how well-structured modulators can be used to obtain efficient parameterizedalgorithms for  Theparameterized complexity ,34 paraA:Width measures Treewidth has become anextremely successful structural parameter with a wide range of applicationsin many fields of computer science. However, treewidth is not suitable foruse in dense graphs. This led to the development of algorithms that use theparameter clique-width  to denote the setA, then for a.The set of natural numbers  will bedenoted by internal nodes. If S is a set of leaves of a tree T,then T(S)denotes the smallest connected subtree spanning S.We will use standard graph theoretic terminology and notation (cf.). All grN(A) the set of neighbors of A in A contains a singlevertex v, we use N(v) instead of V and E as shorthand for V(G) and E(G), respectively, when thegraph is clear from context. Two vertex sets A, B are overlapping if Gobtained by deleting A.Given a graph modulator tohereditary if it is closed under vertex deletion. A graph H is an inducedsubgraph of G if H can be obtained by deleting vertices  from G. ForG[A] to denote the subgraph of G obtained by deleting free graphs is the class ofall graphs which do not contain any graph in obstructions, and we saythat the class of characterizedbyGiven a graph parameterizedproblemx the main part andk the parameter.We refer the reader to the standard textbooks , 16, 34 fixed-parameter tractable(FPT in short) if a given instance  can be solved in timef is an arbitrarycomputable function of k. Algorithms withrunning time in this form are called fixed-parameteralgorithms, and we also slightly abuse notaton and refer to thisruntime simply as FPT time.A parameterized problem paraNP-hard if theunparameterized problem corresponding to the restriction of Minimum Vertex Coverproblem parameterized by the degree of a graph is paraNP-hard, since Minimum Vertex Cover remains NP-hard even on graphsof degree at most 3.A parameterized problem The notions and terminology introduced in this subsection play animportant role in particular when formally defining our parametersSect.  and for split of a connected graphV such that every vertexof frontiers of thesplit.A non-trivial if both sides have at least two vertices. A connectedgraph which does not contain a non-trivial split is called prime. A bipartition is trivialif one of its parts is the empty set or a singleton. Cliques and stars are calleddegenerate graphs; notice that everynon-trivial bipartition of their vertices is a split.A split is said to be split-modules instead of splits wheresuitable. A set split-module ofG if there exists a connected componentG such thatA is asplit-module then A can be partitioned intoV andadjacent if there existx and y are adjacent.Let Our algorithms for computing well-structured modulators will relyon two deep results related to splits, specifically Theorems graph-labeled tree (which canbe viewed as a more modern approach to the previously used split-decompositions) is a pair T is a tree andu of T is labeled by a graph T incident to u andvertices of G(u). When clear from the context, we may use u as a shorthand for V(u) to denote V(G(u)) and we say that an edge of T incident to u is incident to the vertex of G(u) mapped to it. Graph-labeledtrees were introduced by Gioan and Paul ) bipartiteadjacency matrixRank-width was introduced by Oum and Seymour  and is cG andG. Thecut-rank function G is defined asfollows: For a bipartition  of the vertex set X can be seen as a subgraph such thatFor a graph rank-decomposition of a graphG is a pair T is a tree of maximumdegree 3 and e of T, the connected componentsof X, Y) of the set of leavesof T. The width of an edge e of arank-decomposition width of T. The rank-width of G, rw(G) in short, is the minimum width over allrank-decompositions of G. A graph classunbounded rank-width iffor each A An example of a rank-decomposition is provided in Fig. n-vertex graphG, we can output a rank-decomposition ofwidth at most k or confirm that therank-width of G is larger than k in time f is a computablefunction. Let\doHere we introduce monadic second order logic, which will play acrucial role in our positive Sect.  aswell x, y, z, and aninfinite supply of set variables, denoted by uppercase letters X, Y, Z. Formulas ofmonadic second-order logic (MSO) areconstructed from atomic formulas E, X(x), andE expresses adjacency, X(x) expresses that the vertexx is in the set X. From this, we define the semantics of monadic second-order logicin the standard way  Let q, l be non-negative integers.For a graph G and an l-tuple G,we define MSO-q-type of inG.types. Itfollows from Fact Since we will only be dealing with MSO logic, throughout the paperwe will refer to MSO-types simply as q and l benon-negative integer constants. Let G be agraph, and let l-tuple of sets ofvertices of G. One can compute a formulal-tuple Let q aR be a system ofrepresentatives of q and l are constant, we can consider both the cardinality of R and the time required to compute it as constants.Let R is a constant, this can be done in time f) as checking whether Gsatisfied q).Let mentclass2pt{miniml-tuple of subsets ofR is a system ofrepresentatives of Let The remainder of the section introduces the classical notion of MSOgames  Let partial isomorphismbetweenandFor every For every In the definition of MSO games given below, we denote theconcatenation of tuple G and k-tuple of subsets ofV(G) andlet k-tuple of subsets ofq be a non-negativeinteger. The q-round MSO game onGandstarting fromPoint move: The Spoiler picks a vertex ineither G or Set move: The Spoiler picks a subset ofV(G) or a subset of Let q-round game, and let q-round game, so that  underT, soG.We define the mapping k. Given a rankdecomposition e ofX, Y) of the leaves ofe of T. There is a natural bijection T to theedges of A issplit-module either a in A and Bswitching their roles and b taking therole of a.A and Cswitching their roles and c taking therole of a.Since k. We concludethat the width of k and thusWe prove that the width of C, we obtainthe following corollary.By repeating the proof technique of Lemma A, B pairwise disjoint split-modules such thatLet Let k. LetA, B and a, b, respectively. Let mentclass2pt{minimThe following lemma in essence shows that the relation of being in asplit-module of small rank-width is transitive (assuming sufficiently highrank-width). The significance of this will become clear later on.G suchthat G andLet G. Let k.Since rank-width is preserved by taking induced subgraphs, the graphsk. We finish the proof by applying Lemma A, B, and C and a, b, and c, respectively. If entclass1pt{minimaG be a graph andV(G) by letting M of G such that G is clear fromcontext.Let Using Lemma U of G suchthat For every G be a graph andG, soG such thatG containing u, v, w. By Lemma G) suchthat Let v, and let G) such thatG) andNow let G of rank-width atleast k.Every graph Next, we state a simple but useful observation.G. Then Let Now that we know mentclass2pt{minimn-vertex graphG of rank-width at least v, w, we can decide whether f.Let G is connected. By Theorem G is the bipartition of leavesof T induced either by removing an internaltree-edge of T or an edge created by anode-split of a degenerate vertex of T.From Observation mentclass2pt{minimG are leaves ofT and we can find a path P between v andw in T intime linear in size of T. Since the number ofnodes in a split-decomposition is linear in the number of vertices of the originalgraph  is a completegraph. Our first examples deal with two classical NP-hard graph problems,specifically Establishing the following theorem is the main objective of thissubsection.FPT parameterized byLet G, the “only if”direction is immediate; in other words, being polynomial-time tractable onSince MinVC ispolynomial-time tractable on a graph class FPT.If k-well-structured modulator Let FPT time . OtherwiG must be a superset of eitheri in signatures. Each vertex cover Yof G can be associated with at least onesignature f, constructed in the followingway: for each Since for each G and a signature f, we construct a partial vertex cover C of C intersects someZ it must hold that Ceither has rank-width at most k (in the casei), or C is in C does not intersectC has bounded rank-width) orthe polynomial algorithm  in the complementgraph.d is exactly theclass of d leaves. The degree bounds the size of a maximum clique, againresulting in a simple folklore branching algorithm. The class of graphsof degree at most k-well-structured modulators to examples of graph classes which arenot characterized by a finite obstruction set . In the following lemmas, ndenotes the size of the vertex set of the input graph.Next, we turn our attention to computing k-well-structured modulator to the class of forests in timef.It is possible to compute a G is at least k-well-structured modulator can be computed using Courcelle’sTheorem in time at most f. V(G) into equivalenceclasses of f, and sets First, it uses Proposition X, Y, Z) of equivalence classes of S, and restarts the second step withSecond, for each tuple  . If nosIt is easy to verify that the running time of f. As for correctness, let us assume for a contradiction thatH obtainedfrom G after deleting all vertices inelements of C.Clearly, neither C nor any other cycle inH intersects less than 4 equivalenceclasses of We begin by describing our algorithm X of C spans at least 4equivalence classes, H must contain at leasttwo neighbors of X in X ; let us denote these vertices X,the vertices Hwhich spans at most 3 equivalence classes, contradicting our previous conclusionthat no such cycles are present in H. Hencewe may conclude that X.Moreover, assume C intersects each equivalence class at most once. Butthen it must be the case that C also forms acycle in H mustindeed be acyclic.The only case we are left with now is that G contains a minimal k-well-structured modulator G after deleting j.For the other direction, assume that chordless if it is also an induced cycle of length atleast 4, and a graph is chordal if it containsno chordless cycles.For the next result, recall that a cycle is k-well-structured modulator to the class of chordal graphs in timeIt is possible to compute a G is at least k-well-structured modulator can be computed using Courcelle’sTheorem in time at most f. V(G) into equivalenceclasses of First, it uses Proposition X, Y, Z) of equivalence classes of S, and restarts the second step withSecond, for each tuple  Let q and k be non-negative integers,Gand k-well-structured modulators q-similar if all of the following conditions are met:There exists an isomorphism v isadjacent to For every If For each q and k be non-negative integers, Gand k-well-structured modulators q-similar, then it holds that Let q-round MSO gameplayed on W and assume without loss of generality that Suppose Spoiler makes a set move s and again assume without loss of generality thatSuppose Spoiler makes a point move Assume Duplicator plays according to this strategy and consider aplay of the q-round MSO game on G and V(G) and V(G) and Let Let Let G. Fromthe correctness of G.In the first case we have Let Let Let For Next, we show that small representatives can be computedefficiently.f, g such that one can compute,for an input graph G of rank-width at mostk and Let T of intime T. By Fact q-types isfinite for each q, so we can think of the totalnumber of checks and the size of each checked graph T and the size of the graphBy Lemma mentclass2pt{minimFinally, in Lemma q be a non-negative integerconstant and G and a k-well-structuredmodulator G into fsuch that one can in time k-well-structuredmodulator q-similar and for eachLet q-type asf. Now letWe construct a disjoint union of G, weadd edges between every If v and every for every It is easy to verify that q-similar.For G be a graph,q be the nesting depth ofquantifiers in k-well-structured modulatorto G andq-similar, it followsfrom Lemma Let For completeness, we remark that the same proof can be used toobtain analogues of Theorem mentclasspt{minimac-Coloring asks whether thevertices of an input graph G can be colored byc colors so that each pair of neighbors havedistinct colors. From the connection between c-Coloring, its generalizationList c-Coloring and modulators [List c-Coloring [We conclude the section by showcasing an example application ofTheorem Coloring , Page 5,Coloring parameterized byFPT.For each Instance: A graph G and an integer Question: Is there a set We say that dominating set if everyvertex in G either is in S or has a neighbor in S. We willneed the following lemma before we proceed to the main result of this secton.In the wake of Theorem p-cardinality dominating set in a graph G having a k-cardinalitymodulator The problem of finding a D. We branch over the at most q in q to D. If G, we discard the current branch; hence this procedureproduces a total of at most Let X. For P of all such chosen representatives. We thenbranch over all subsets Q of Q into D. Since D is a dominating set, and output the minimum dominating setobtained in this manner.Now consider a branch where p, then thealgorithm will find such a set. So, assume there exists a p-cardinality dominating set G. Consider the brancharising from the first branching rule obtained as follows. Let L chosen by the algorithm, and consider the branch where anarbitrary D. Hence,after the first branching, there is a branch where i-th step ofthe first branching. If P. SinceLand X as G. Furthermore, It is easily observed from the description that the running timeis FPT. For correctness, from the final check it follows that any set outputed bythe algorithm will be a dominating set. It remains to show that if there exists adominating set of cardinality FPT parameterized byparaNP-hard parameterizedby There exists an MSO formula dom(S)express that S is a dominating set in G, and let cyc(S) express that S intersects every To prove Theorem k be the cardinality of amodulator in G to k in G to G. Letr andk, and if G must intersect X and hence setting To argue that the above claim holds, let A of cardinality at most r whichintersects every B in G of cardinality at most r; this can also be done in FPT time by Lemma So, assume A or B exists, then we output YES and otherwise we output NO.Hence the claim is indeed true.Finally, if Dominating Set problem, which takesas input a graph G and an integer j and asks to find a dominating set of size at mostj, is NP-hard on G, j) be aDominatingSet of degree at most 3. Then we construct G by addingq adjacent toevery vertex of every such q and an arbitraryvertex of G. It is easy to check thatWe proceed by arguing that this claim is also correct. It is knownthat the t most 3   is a YES-instance of Dominating Set if and only if D in G ofcardinality j. Then the set We claim that  is a YES-instance of Dominating Set, so assume this is not the case. But thenG is also a dominating setof j in V(G). Consequently,  is a YES-instance of Dominating Set and the claim holds.On the other hand, assume there exists a set The theorem now follows from the two claims provedabove.Minimum VertexCover and Maximum Clique(Theorem We have introduced a family of structural parameters which push thefrontiers of fixed-parameter tractability beyond rank-width and modulator size for awide range of problems. In particular, the well-structure number can be computedefficiently Theorem  and usedTheorem  as well Theorem . We remaX can be seen as asubgraph such that For future work, it would be interesting to see whether the notionof split-modules introduced in this work can be naturally generalized. Inparticular, a split-module kernelization [Finally, we remark that well-structured modulators have also foundapplications in the area of data reduction and lization . Inpartconstant .


In this paper, we mainly consider the stochastic second-order cone complementarity problem (SSOCCP). Due to the existence of stochastic variable, the SSOCCP may have no solutions. In order to deal with this problem, we first regard the merit function of the stochastic second-order cone complementarity problem as the loss function and then present a low-risk deterministic model that is a conditional value-at-risk (CVaR) model. However, there may be two difficulties for solving the CVaR model directly: One is that the objective function is a non-smoothing function. The other is that the objective function contains expectation.  In view of these two problems, we present the approximation problems of the model by using a smoothing method and a sample average approximation technique. Furthermore, we give the convergence results of global optimal solutions and the convergence results of stationary points of the approximation problems, respectively. The second-order cone in Second-order cone complementarity problem (SOCCP) is as follows: Find a vector The KKT condition of second-order cone programming (SOCP) problem can be equivalent to a second-order cone complementarity problem. So, many researchers pay attention to considering SOCCP.x. “a.s.” is the abbreviation of “almost surely”. Because of the existence of stochastic factor ω, generally there is no vector x satisfying  second-order cone complementarity function, that is, . And by ,2.1\docIn this paper, we consider a Fischer–Burimister(FB) second-order cone complementarity function, taking advantage of the knowledge about Jordan algebra, If the mapping In fact, we have Let τ-Based on the definition of ction in , we presτ-A function τ-A function τ-equi-coerciveness.Similar to the definition of equi-coerciveness in  , we givτ-equi-coercive on A function x and a parameter u, its probability is defined asω. For a given confidence level α, the definition of x is given as follows:Before giving a CVaR model, we give the definition of Value-at-Risk (VaR). For a variable ferences , 18, Rocα, CVaR is defined as the conditional expectation of the loss associated with x relative to that loss being E denotes mathematics expectation. From references . From , 5.11),,\documenx, we get that Next, we will prove that }μ≠0, by , we know. Taking  into accx, we obtain that Noting that Ω is a compact set and In this paper, we consider a stochastic second-order cone complementarity problem. The boundedness of level sets of  problem  are boun, SSOCCP  may have