| """ |
| 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 |
|
|
|
|
| @nx._dispatchable( |
| node_attrs="demand", edge_attrs={"capacity": float("inf"), "weight": 0} |
| ) |
| 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] |
|
|
|
|
| @nx._dispatchable( |
| node_attrs="demand", edge_attrs={"capacity": float("inf"), "weight": 0} |
| ) |
| 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) |
| >>> flowDict |
| {'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}} |
| """ |
| return nx.network_simplex(G, demand=demand, capacity=capacity, weight=weight)[1] |
|
|
|
|
| @nx._dispatchable(edge_attrs={"weight": 0}) |
| 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). |
| |
| Examples |
| -------- |
| >>> 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) |
| >>> flowDict |
| {'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}} |
| >>> nx.cost_of_flow(G, flowDict) |
| 24 |
| """ |
| return sum((flowDict[u][v] * d.get(weight, 0) for u, v, d in G.edges(data=True))) |
|
|
|
|
| @nx._dispatchable(edge_attrs={"capacity": float("inf"), "weight": 0}) |
| 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) |
|
|
