Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

Source code for networkx.algorithms.flow.mincost

"""
Minimum cost flow algorithms on directed connected graphs.
"""

__all__ = ["min_cost_flow_cost", "min_cost_flow", "cost_of_flow", "max_flow_min_cost"]

import networkx as nx


[docs]def min_cost_flow_cost(G, demand="demand", capacity="capacity", weight="weight"): r"""Find the cost of a minimum cost flow satisfying all demands in digraph G. G is a digraph with edge costs and capacities and in which nodes 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. Parameters ---------- G : NetworkX graph DiGraph on which a minimum cost flow satisfying all demands is to be found. demand : string Nodes of the graph G are expected to have an attribute demand 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 demand. Default value: 'demand'. capacity : string Edges of the graph G are expected to have an attribute capacity that indicates how much flow the edge can support. If this attribute is not present, the edge is considered to have infinite capacity. Default value: 'capacity'. weight : string Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered to be 0. Default value: 'weight'. Returns ------- flowCost : integer, float Cost of a minimum cost flow satisfying all demands. Raises ------ NetworkXError This exception is raised if the input graph is not directed or not connected. NetworkXUnfeasible This exception is raised in the following situations: * The sum of the demands is not zero. Then, there is no flow satisfying all demands. * There is no flow satisfying all demand. NetworkXUnbounded This exception is raised if the digraph G has a cycle of negative cost and infinite capacity. Then, the cost of a flow satisfying all demands is unbounded below. See also -------- cost_of_flow, max_flow_min_cost, min_cost_flow, 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). Examples -------- A simple example of a min cost flow problem. >>> G = nx.DiGraph() >>> G.add_node("a", demand=-5) >>> G.add_node("d", demand=5) >>> G.add_edge("a", "b", weight=3, capacity=4) >>> G.add_edge("a", "c", weight=6, capacity=10) >>> G.add_edge("b", "d", weight=1, capacity=9) >>> G.add_edge("c", "d", weight=2, capacity=5) >>> flowCost = nx.min_cost_flow_cost(G) >>> flowCost 24 """ return nx.network_simplex(G, demand=demand, capacity=capacity, weight=weight)[0]
[docs]def min_cost_flow(G, demand="demand", capacity="capacity", weight="weight"): r"""Returns a minimum cost flow satisfying all demands in digraph G. G is a digraph with edge costs and capacities and in which nodes 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. Parameters ---------- G : NetworkX graph DiGraph on which a minimum cost flow satisfying all demands is to be found. demand : string Nodes of the graph G are expected to have an attribute demand 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 demand. Default value: 'demand'. capacity : string Edges of the graph G are expected to have an attribute capacity that indicates how much flow the edge can support. If this attribute is not present, the edge is considered to have infinite capacity. Default value: 'capacity'. weight : string Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered to be 0. Default value: 'weight'. Returns ------- flowDict : dictionary Dictionary of dictionaries keyed by nodes such that flowDict[u][v] is the flow edge (u, v). Raises ------ NetworkXError This exception is raised if the input graph is not directed or not connected. NetworkXUnfeasible This exception is raised in the following situations: * The sum of the demands is not zero. Then, there is no flow satisfying all demands. * There is no flow satisfying all demand. NetworkXUnbounded This exception is raised if the digraph G has a cycle of negative cost and infinite capacity. Then, the cost of a flow satisfying all demands is unbounded below. See also -------- cost_of_flow, max_flow_min_cost, 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). Examples -------- A simple example of a min cost flow problem. >>> G = nx.DiGraph() >>> G.add_node("a", demand=-5) >>> G.add_node("d", demand=5) >>> G.add_edge("a", "b", weight=3, capacity=4) >>> G.add_edge("a", "c", weight=6, capacity=10) >>> G.add_edge("b", "d", weight=1, capacity=9) >>> G.add_edge("c", "d", weight=2, capacity=5) >>> flowDict = nx.min_cost_flow(G) """ return nx.network_simplex(G, demand=demand, capacity=capacity, weight=weight)[1]
[docs]def cost_of_flow(G, flowDict, weight="weight"): """Compute the cost of the flow given by flowDict on graph G. Note that this function does not check for the validity of the flow flowDict. This function will fail if the graph G and the flow don't have the same edge set. Parameters ---------- G : NetworkX graph DiGraph on which a minimum cost flow satisfying all demands is to be found. weight : string Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered to be 0. Default value: 'weight'. flowDict : dictionary Dictionary of dictionaries keyed by nodes such that flowDict[u][v] is the flow edge (u, v). Returns ------- cost : Integer, float The total cost of the flow. This is given by the sum over all edges of the product of the edge's flow and the edge's weight. See also -------- max_flow_min_cost, 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). """ return sum((flowDict[u][v] * d.get(weight, 0) for u, v, d in G.edges(data=True)))
[docs]def max_flow_min_cost(G, s, t, capacity="capacity", weight="weight"): """Returns a maximum (s, t)-flow of minimum cost. G is a digraph with edge costs and capacities. There is a source node s and a sink node t. This function finds a maximum flow from s to t whose total cost is minimized. Parameters ---------- G : NetworkX graph DiGraph on which a minimum cost flow satisfying all demands is to be found. s: node label Source of the flow. t: node label Destination of the flow. capacity: string Edges of the graph G are expected to have an attribute capacity that indicates how much flow the edge can support. If this attribute is not present, the edge is considered to have infinite capacity. Default value: 'capacity'. weight: string Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered to be 0. Default value: 'weight'. Returns ------- flowDict: dictionary Dictionary of dictionaries keyed by nodes such that flowDict[u][v] is the flow edge (u, v). Raises ------ NetworkXError This exception is raised if the input graph is not directed or not connected. NetworkXUnbounded This exception is raised if there is an infinite capacity path from s to t in G. In this case there is no maximum flow. This 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 -------- 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). Examples -------- >>> G = nx.DiGraph() >>> G.add_edges_from( ... [ ... (1, 2, {"capacity": 12, "weight": 4}), ... (1, 3, {"capacity": 20, "weight": 6}), ... (2, 3, {"capacity": 6, "weight": -3}), ... (2, 6, {"capacity": 14, "weight": 1}), ... (3, 4, {"weight": 9}), ... (3, 5, {"capacity": 10, "weight": 5}), ... (4, 2, {"capacity": 19, "weight": 13}), ... (4, 5, {"capacity": 4, "weight": 0}), ... (5, 7, {"capacity": 28, "weight": 2}), ... (6, 5, {"capacity": 11, "weight": 1}), ... (6, 7, {"weight": 8}), ... (7, 4, {"capacity": 6, "weight": 6}), ... ] ... ) >>> mincostFlow = nx.max_flow_min_cost(G, 1, 7) >>> mincost = nx.cost_of_flow(G, mincostFlow) >>> mincost 373 >>> from networkx.algorithms.flow import maximum_flow >>> maxFlow = maximum_flow(G, 1, 7)[1] >>> nx.cost_of_flow(G, maxFlow) >= mincost True >>> mincostFlowValue = sum((mincostFlow[u][7] for u in G.predecessors(7))) - sum( ... (mincostFlow[7][v] for v in G.successors(7)) ... ) >>> mincostFlowValue == nx.maximum_flow_value(G, 1, 7) True """ maxFlow = nx.maximum_flow_value(G, s, t, capacity=capacity) H = nx.DiGraph(G) H.add_node(s, demand=-maxFlow) H.add_node(t, demand=maxFlow) return min_cost_flow(H, capacity=capacity, weight=weight)