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""" |
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Minimum cost flow algorithms on directed connected graphs. |
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""" |
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__all__ = ["min_cost_flow_cost", "min_cost_flow", "cost_of_flow", "max_flow_min_cost"] |
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import networkx as nx |
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@nx._dispatchable( |
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node_attrs="demand", edge_attrs={"capacity": float("inf"), "weight": 0} |
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) |
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def min_cost_flow_cost(G, demand="demand", capacity="capacity", weight="weight"): |
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r"""Find the cost of a minimum cost flow satisfying all demands in digraph G. |
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G is a digraph with edge costs and capacities and in which nodes |
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have demand, i.e., they want to send or receive some amount of |
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flow. A negative demand means that the node wants to send flow, a |
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positive demand means that the node want to receive flow. A flow on |
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the digraph G satisfies all demand if the net flow into each node |
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is equal to the demand of that node. |
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Parameters |
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---------- |
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G : NetworkX graph |
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DiGraph on which a minimum cost flow satisfying all demands is |
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to be found. |
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demand : string |
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Nodes of the graph G are expected to have an attribute demand |
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that indicates how much flow a node wants to send (negative |
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demand) or receive (positive demand). Note that the sum of the |
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demands should be 0 otherwise the problem in not feasible. If |
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this attribute is not present, a node is considered to have 0 |
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demand. Default value: 'demand'. |
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capacity : string |
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Edges of the graph G are expected to have an attribute capacity |
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that indicates how much flow the edge can support. If this |
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attribute is not present, the edge is considered to have |
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infinite capacity. Default value: 'capacity'. |
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weight : string |
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Edges of the graph G are expected to have an attribute weight |
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that indicates the cost incurred by sending one unit of flow on |
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that edge. If not present, the weight is considered to be 0. |
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Default value: 'weight'. |
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Returns |
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------- |
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flowCost : integer, float |
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Cost of a minimum cost flow satisfying all demands. |
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Raises |
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------ |
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NetworkXError |
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This exception is raised if the input graph is not directed or |
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not connected. |
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NetworkXUnfeasible |
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This exception is raised in the following situations: |
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* The sum of the demands is not zero. Then, there is no |
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flow satisfying all demands. |
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* There is no flow satisfying all demand. |
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NetworkXUnbounded |
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This exception is raised if the digraph G has a cycle of |
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negative cost and infinite capacity. Then, the cost of a flow |
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satisfying all demands is unbounded below. |
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See also |
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-------- |
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cost_of_flow, max_flow_min_cost, min_cost_flow, network_simplex |
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Notes |
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----- |
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This algorithm is not guaranteed to work if edge weights or demands |
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are floating point numbers (overflows and roundoff errors can |
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cause problems). As a workaround you can use integer numbers by |
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multiplying the relevant edge attributes by a convenient |
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constant factor (eg 100). |
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Examples |
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-------- |
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A simple example of a min cost flow problem. |
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>>> G = nx.DiGraph() |
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>>> G.add_node("a", demand=-5) |
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>>> G.add_node("d", demand=5) |
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>>> G.add_edge("a", "b", weight=3, capacity=4) |
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>>> G.add_edge("a", "c", weight=6, capacity=10) |
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>>> G.add_edge("b", "d", weight=1, capacity=9) |
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>>> G.add_edge("c", "d", weight=2, capacity=5) |
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>>> flowCost = nx.min_cost_flow_cost(G) |
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>>> flowCost |
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24 |
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""" |
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return nx.network_simplex(G, demand=demand, capacity=capacity, weight=weight)[0] |
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@nx._dispatchable( |
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node_attrs="demand", edge_attrs={"capacity": float("inf"), "weight": 0} |
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) |
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def min_cost_flow(G, demand="demand", capacity="capacity", weight="weight"): |
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r"""Returns a minimum cost flow satisfying all demands in digraph G. |
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|
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G is a digraph with edge costs and capacities and in which nodes |
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have demand, i.e., they want to send or receive some amount of |
|
|
flow. A negative demand means that the node wants to send flow, a |
|
|
positive demand means that the node want to receive flow. A flow on |
|
|
the digraph G satisfies all demand if the net flow into each node |
|
|
is equal to the demand of that node. |
|
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|
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|
Parameters |
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---------- |
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G : NetworkX graph |
|
|
DiGraph on which a minimum cost flow satisfying all demands is |
|
|
to be found. |
|
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|
|
|
demand : string |
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Nodes of the graph G are expected to have an attribute demand |
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|
that indicates how much flow a node wants to send (negative |
|
|
demand) or receive (positive demand). Note that the sum of the |
|
|
demands should be 0 otherwise the problem in not feasible. If |
|
|
this attribute is not present, a node is considered to have 0 |
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demand. Default value: 'demand'. |
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capacity : string |
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Edges of the graph G are expected to have an attribute capacity |
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that indicates how much flow the edge can support. If this |
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attribute is not present, the edge is considered to have |
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infinite capacity. Default value: 'capacity'. |
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weight : string |
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Edges of the graph G are expected to have an attribute weight |
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that indicates the cost incurred by sending one unit of flow on |
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that edge. If not present, the weight is considered to be 0. |
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Default value: 'weight'. |
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Returns |
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------- |
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flowDict : dictionary |
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Dictionary of dictionaries keyed by nodes such that |
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flowDict[u][v] is the flow edge (u, v). |
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Raises |
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------ |
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NetworkXError |
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This exception is raised if the input graph is not directed or |
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|
not connected. |
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NetworkXUnfeasible |
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This exception is raised in the following situations: |
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* The sum of the demands is not zero. Then, there is no |
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flow satisfying all demands. |
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* There is no flow satisfying all demand. |
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NetworkXUnbounded |
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This exception is raised if the digraph G has a cycle of |
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negative cost and infinite capacity. Then, the cost of a flow |
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satisfying all demands is unbounded below. |
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|
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See also |
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-------- |
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cost_of_flow, max_flow_min_cost, min_cost_flow_cost, network_simplex |
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|
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Notes |
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----- |
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This algorithm is not guaranteed to work if edge weights or demands |
|
|
are floating point numbers (overflows and roundoff errors can |
|
|
cause problems). As a workaround you can use integer numbers by |
|
|
multiplying the relevant edge attributes by a convenient |
|
|
constant factor (eg 100). |
|
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|
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|
Examples |
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-------- |
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|
A simple example of a min cost flow problem. |
|
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|
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|
>>> G = nx.DiGraph() |
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>>> G.add_node("a", demand=-5) |
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>>> G.add_node("d", demand=5) |
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>>> G.add_edge("a", "b", weight=3, capacity=4) |
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>>> G.add_edge("a", "c", weight=6, capacity=10) |
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>>> G.add_edge("b", "d", weight=1, capacity=9) |
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>>> G.add_edge("c", "d", weight=2, capacity=5) |
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>>> flowDict = nx.min_cost_flow(G) |
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>>> flowDict |
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{'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}} |
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""" |
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return nx.network_simplex(G, demand=demand, capacity=capacity, weight=weight)[1] |
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@nx._dispatchable(edge_attrs={"weight": 0}) |
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def cost_of_flow(G, flowDict, weight="weight"): |
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"""Compute the cost of the flow given by flowDict on graph G. |
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Note that this function does not check for the validity of the |
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flow flowDict. This function will fail if the graph G and the |
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flow don't have the same edge set. |
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Parameters |
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---------- |
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G : NetworkX graph |
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|
DiGraph on which a minimum cost flow satisfying all demands is |
|
|
to be found. |
|
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weight : string |
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Edges of the graph G are expected to have an attribute weight |
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|
that indicates the cost incurred by sending one unit of flow on |
|
|
that edge. If not present, the weight is considered to be 0. |
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Default value: 'weight'. |
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flowDict : dictionary |
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Dictionary of dictionaries keyed by nodes such that |
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flowDict[u][v] is the flow edge (u, v). |
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Returns |
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------- |
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cost : Integer, float |
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The total cost of the flow. This is given by the sum over all |
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edges of the product of the edge's flow and the edge's weight. |
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See also |
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-------- |
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max_flow_min_cost, min_cost_flow, min_cost_flow_cost, network_simplex |
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|
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Notes |
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----- |
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This algorithm is not guaranteed to work if edge weights or demands |
|
|
are floating point numbers (overflows and roundoff errors can |
|
|
cause problems). As a workaround you can use integer numbers by |
|
|
multiplying the relevant edge attributes by a convenient |
|
|
constant factor (eg 100). |
|
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|
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Examples |
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-------- |
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>>> G = nx.DiGraph() |
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>>> G.add_node("a", demand=-5) |
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>>> G.add_node("d", demand=5) |
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>>> G.add_edge("a", "b", weight=3, capacity=4) |
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>>> G.add_edge("a", "c", weight=6, capacity=10) |
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>>> G.add_edge("b", "d", weight=1, capacity=9) |
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>>> G.add_edge("c", "d", weight=2, capacity=5) |
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>>> flowDict = nx.min_cost_flow(G) |
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>>> flowDict |
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{'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}} |
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>>> nx.cost_of_flow(G, flowDict) |
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24 |
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""" |
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return sum((flowDict[u][v] * d.get(weight, 0) for u, v, d in G.edges(data=True))) |
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@nx._dispatchable(edge_attrs={"capacity": float("inf"), "weight": 0}) |
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def max_flow_min_cost(G, s, t, capacity="capacity", weight="weight"): |
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"""Returns a maximum (s, t)-flow of minimum cost. |
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G is a digraph with edge costs and capacities. There is a source |
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node s and a sink node t. This function finds a maximum flow from |
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s to t whose total cost is minimized. |
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Parameters |
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---------- |
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G : NetworkX graph |
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DiGraph on which a minimum cost flow satisfying all demands is |
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to be found. |
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s: node label |
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Source of the flow. |
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t: node label |
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Destination of the flow. |
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capacity: string |
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Edges of the graph G are expected to have an attribute capacity |
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that indicates how much flow the edge can support. If this |
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attribute is not present, the edge is considered to have |
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infinite capacity. Default value: 'capacity'. |
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weight: string |
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Edges of the graph G are expected to have an attribute weight |
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that indicates the cost incurred by sending one unit of flow on |
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that edge. If not present, the weight is considered to be 0. |
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Default value: 'weight'. |
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Returns |
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------- |
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flowDict: dictionary |
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|
Dictionary of dictionaries keyed by nodes such that |
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flowDict[u][v] is the flow edge (u, v). |
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|
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Raises |
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------ |
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NetworkXError |
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This exception is raised if the input graph is not directed or |
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|
not connected. |
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|
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NetworkXUnbounded |
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This exception is raised if there is an infinite capacity path |
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from s to t in G. In this case there is no maximum flow. This |
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exception is also raised if the digraph G has a cycle of |
|
|
negative cost and infinite capacity. Then, the cost of a flow |
|
|
is unbounded below. |
|
|
|
|
|
See also |
|
|
-------- |
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|
cost_of_flow, min_cost_flow, min_cost_flow_cost, network_simplex |
|
|
|
|
|
Notes |
|
|
----- |
|
|
This algorithm is not guaranteed to work if edge weights or demands |
|
|
are floating point numbers (overflows and roundoff errors can |
|
|
cause problems). As a workaround you can use integer numbers by |
|
|
multiplying the relevant edge attributes by a convenient |
|
|
constant factor (eg 100). |
|
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|
|
|
Examples |
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|
-------- |
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>>> G = nx.DiGraph() |
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>>> G.add_edges_from( |
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... [ |
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... (1, 2, {"capacity": 12, "weight": 4}), |
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... (1, 3, {"capacity": 20, "weight": 6}), |
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... (2, 3, {"capacity": 6, "weight": -3}), |
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... (2, 6, {"capacity": 14, "weight": 1}), |
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... (3, 4, {"weight": 9}), |
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... (3, 5, {"capacity": 10, "weight": 5}), |
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... (4, 2, {"capacity": 19, "weight": 13}), |
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... (4, 5, {"capacity": 4, "weight": 0}), |
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... (5, 7, {"capacity": 28, "weight": 2}), |
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... (6, 5, {"capacity": 11, "weight": 1}), |
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... (6, 7, {"weight": 8}), |
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... (7, 4, {"capacity": 6, "weight": 6}), |
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... ] |
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... ) |
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>>> mincostFlow = nx.max_flow_min_cost(G, 1, 7) |
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>>> mincost = nx.cost_of_flow(G, mincostFlow) |
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>>> mincost |
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373 |
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>>> from networkx.algorithms.flow import maximum_flow |
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>>> maxFlow = maximum_flow(G, 1, 7)[1] |
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>>> nx.cost_of_flow(G, maxFlow) >= mincost |
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True |
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>>> mincostFlowValue = sum((mincostFlow[u][7] for u in G.predecessors(7))) - sum( |
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... (mincostFlow[7][v] for v in G.successors(7)) |
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... ) |
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>>> mincostFlowValue == nx.maximum_flow_value(G, 1, 7) |
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True |
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""" |
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maxFlow = nx.maximum_flow_value(G, s, t, capacity=capacity) |
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H = nx.DiGraph(G) |
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H.add_node(s, demand=-maxFlow) |
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H.add_node(t, demand=maxFlow) |
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return min_cost_flow(H, capacity=capacity, weight=weight) |
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|
