# networkx.algorithms.flow.maximum_flow¶

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

Find a maximum single-commodity flow.

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.

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.

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.

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.

Examples

>>> import networkx as nx
>>> G = nx.DiGraph()


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