Find a minimum cost flow satisfying all demands in digraph G.
This is a primal network simplex algorithm that uses the leaving arc rule to prevent cycling.
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
demand: string :
capacity: string :
weight: string :


Returns :  flowCost: integer, float :
flowDict: dictionary :

Raises :  NetworkXError :
NetworkXUnfeasible :
NetworkXUnbounded :

See also
cost_of_flow, max_flow_min_cost, min_cost_flow, min_cost_flow_cost
Notes
This algorithm is not guaranteed to work if edge weights are floating point numbers (overflows and roundoff errors can cause problems).
References
W. J. Cook, W. H. Cunningham, W. R. Pulleyblank and A. Schrijver. Combinatorial Optimization. WileyInterscience, 1998.
Examples
A simple example of a min cost flow problem.
>>> import networkx as nx
>>> 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, flowDict = nx.network_simplex(G)
>>> flowCost
24
>>> flowDict
{'a': {'c': 1, 'b': 4}, 'c': {'d': 1}, 'b': {'d': 4}, 'd': {}}
The mincost flow algorithm can also be used to solve shortest path problems. To find the shortest path between two nodes u and v, give all edges an infinite capacity, give node u a demand of 1 and node v a demand a 1. Then run the network simplex. The value of a min cost flow will be the distance between u and v and edges carrying positive flow will indicate the path.
>>> G=nx.DiGraph()
>>> G.add_weighted_edges_from([('s','u',10), ('s','x',5),
... ('u','v',1), ('u','x',2),
... ('v','y',1), ('x','u',3),
... ('x','v',5), ('x','y',2),
... ('y','s',7), ('y','v',6)])
>>> G.add_node('s', demand = 1)
>>> G.add_node('v', demand = 1)
>>> flowCost, flowDict = nx.network_simplex(G)
>>> flowCost == nx.shortest_path_length(G, 's', 'v', weight = 'weight')
True
>>> [(u, v) for u in flowDict for v in flowDict[u] if flowDict[u][v] > 0]
[('x', 'u'), ('s', 'x'), ('u', 'v')]
>>> nx.shortest_path(G, 's', 'v', weight = 'weight')
['s', 'x', 'u', 'v']
It is possible to change the name of the attributes used for the algorithm.
>>> G = nx.DiGraph()
>>> G.add_node('p', spam = 4)
>>> G.add_node('q', spam = 2)
>>> G.add_node('a', spam = 2)
>>> G.add_node('d', spam = 1)
>>> G.add_node('t', spam = 2)
>>> G.add_node('w', spam = 3)
>>> G.add_edge('p', 'q', cost = 7, vacancies = 5)
>>> G.add_edge('p', 'a', cost = 1, vacancies = 4)
>>> G.add_edge('q', 'd', cost = 2, vacancies = 3)
>>> G.add_edge('t', 'q', cost = 1, vacancies = 2)
>>> G.add_edge('a', 't', cost = 2, vacancies = 4)
>>> G.add_edge('d', 'w', cost = 3, vacancies = 4)
>>> G.add_edge('t', 'w', cost = 4, vacancies = 1)
>>> flowCost, flowDict = nx.network_simplex(G, demand = 'spam',
... capacity = 'vacancies',
... weight = 'cost')
>>> flowCost
37
>>> flowDict
{'a': {'t': 4}, 'd': {'w': 2}, 'q': {'d': 1}, 'p': {'q': 2, 'a': 2}, 't': {'q': 1, 'w': 1}, 'w': {}}