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maximum_flow(flowG, _s, _t, capacity='capacity', flow_func=None, **kwargs)

Notes

The function used in the flow_func parameter has to return a residual network that follows NetworkX conventions:

The residual network R from an input graph G has the same nodes as G . R is a DiGraph that contains a pair of edges (u, v) and (v, u) iff (u, v) is not a self-loop, and at least one of (u, v) and (v, u) exists in G .

For each edge (u, v) in R , R[u][v]['capacity'] is equal to the capacity of (u, v) in G if it exists in G or zero otherwise. If the capacity is infinite, R[u][v]['capacity'] will have a high arbitrary finite value that does not affect the solution of the problem. This value is stored in R.graph['inf'] . For each edge (u, v) in R , R[u][v]['flow'] represents the flow function of (u, v) and satisfies R[u][v]['flow'] == -R[v][u]['flow'] .

The flow value, defined as the total flow into t , the sink, is stored in R.graph['flow_value'] . Reachability to t using only edges (u, v) such that R[u][v]['flow'] < R[u][v]['capacity'] induces a minimum s - t cut.

Specific algorithms may store extra data in R .

The function should supports an optional boolean parameter value_only. When True, it can optionally terminate the algorithm as soon as the maximum flow value and the minimum cut can be determined.

Parameters

flowG : NetworkX graph: Edges of the graph are expected to have an attribute called 'capacity'. If this attribute is not present, the edge is considered to have infinite capacity.
_s : node: Source node for the flow.
_t : node: Sink node for 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'.
flow_func : function: A function for computing the maximum flow among a pair of nodes in a capacitated graph. The function has to accept at least three parameters: a Graph or Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see Notes). If flow_func is None, the default maximum flow function ( preflow_push ) is used. See below for alternative algorithms. The choice of the default function may change from version to version and should not be relied on. Default value: None.
kwargs : Any other keyword parameter is passed to the function that: computes the maximum flow.

Raises

NetworkXError: The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an instance of one of these two classes, a NetworkXError is raised.
NetworkXUnbounded: If the graph has a path of infinite capacity, the value of a feasible flow on the graph is unbounded above and the function raises a NetworkXUnbounded.

Returns

flow_value : integer, float: Value of the maximum flow, i.e., net outflow from the source.
flow_dict : dict: A dictionary containing the value of the flow that went through each edge.

Find a maximum single-commodity flow.

Examples

>>> G = nx.DiGraph()
... G.add_edge("x", "a", capacity=3.0)
... G.add_edge("x", "b", capacity=1.0)
... G.add_edge("a", "c", capacity=3.0)
... G.add_edge("b", "c", capacity=5.0)
... G.add_edge("b", "d", capacity=4.0)
... G.add_edge("d", "e", capacity=2.0)
... G.add_edge("c", "y", capacity=2.0)
... G.add_edge("e", "y", capacity=3.0)

maximum_flow returns both the value of the maximum flow and a dictionary with all flows.

>>> flow_value, flow_dict = nx.maximum_flow(G, "x", "y")
... flow_value
3.0

>>> print(flow_dict["x"]["b"])
1.0

You can also use alternative algorithms for computing the maximum flow by using the flow_func parameter.

>>> from networkx.algorithms.flow import shortest_augmenting_path
... flow_value == nx.maximum_flow(G, "x", "y", flow_func=shortest_augmenting_path)[
...     0
... ]
True

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

networkx.algorithms.connectivity.cuts.minimum_node_cut networkx.algorithms.flow.edmondskarp.edmonds_karp networkx.algorithms.connectivity.connectivity.local_node_connectivity networkx.algorithms.flow.dinitz_alg.dinitz networkx.algorithms.flow.mincost.max_flow_min_cost networkx.algorithms.connectivity.cuts.minimum_st_edge_cut networkx.algorithms.flow.maxflow.minimum_cut_value networkx.algorithms.connectivity.connectivity.local_edge_connectivity networkx.algorithms.connectivity.connectivity.edge_connectivity networkx.algorithms.flow.maxflow.maximum_flow networkx.algorithms.flow.maxflow.maximum_flow_value networkx.algorithms.connectivity.connectivity.all_pairs_node_connectivity networkx.algorithms.flow.preflowpush.preflow_push networkx.algorithms.connectivity.cuts.minimum_edge_cut networkx.algorithms.flow.gomory_hu.gomory_hu_tree networkx.algorithms.connectivity.connectivity.node_connectivity networkx.algorithms.flow.shortestaugmentingpath.shortest_augmenting_path networkx.algorithms.connectivity.disjoint_paths.node_disjoint_paths networkx.algorithms.flow.boykovkolmogorov.boykov_kolmogorov networkx.algorithms.flow.maxflow.minimum_cut networkx.algorithms.connectivity.connectivity.average_node_connectivity networkx.algorithms.connectivity.disjoint_paths.edge_disjoint_paths networkx.algorithms.connectivity.cuts.minimum_st_node_cut

Local connectivity graph

Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.

Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)

SVG is more flexible but power hungry; and does not scale well to 50 + nodes.

All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them

GitHub : /networkx/algorithms/flow/maxflow.py#27
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