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# max_flow_min_cost

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