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

Use the max-flow min-cut theorem, i.e., the capacity of a minimum capacity cut is equal to the flow value of a maximum flow.

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

NetworkXUnbounded

If the graph has a path of infinite capacity, all cuts have infinite capacity and the function raises a NetworkXError.

Returns

cut_value : integer, float

Value of the minimum cut.

partition : pair of node sets

A partitioning of the nodes that defines a minimum cut.

Compute the value and the node partition of a minimum (s, t)-cut.

See Also

edmonds_karp

meth

maximum_flow

meth

maximum_flow_value

meth

minimum_cut_value

meth

preflow_push

meth

shortest_augmenting_path

meth

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)

minimum_cut computes both the value of the minimum cut and the node partition:

>>> cut_value, partition = nx.minimum_cut(G, "x", "y")
... reachable, non_reachable = partition

'partition' here is a tuple with the two sets of nodes that define the minimum cut. You can compute the cut set of edges that induce the minimum cut as follows:

>>> cutset = set()
... for u, nbrs in ((n, G[n]) for n in reachable):
...  cutset.update((u, v) for v in nbrs if v in non_reachable)
... print(sorted(cutset)) [('c', 'y'), ('x', 'b')]
>>> cut_value == sum(G.edges[u, v]["capacity"] for (u, v) in cutset)
True

You can also use alternative algorithms for computing the minimum cut by using the flow_func parameter.

>>> from networkx.algorithms.flow import shortest_augmenting_path
... cut_value == nx.minimum_cut(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_st_edge_cut networkx.algorithms.connectivity.cuts.minimum_node_cut networkx.algorithms.flow.preflowpush.preflow_push networkx.algorithms.flow.edmondskarp.edmonds_karp networkx.algorithms.flow.boykovkolmogorov.boykov_kolmogorov networkx.algorithms.flow.maxflow.maximum_flow networkx.algorithms.flow.maxflow.minimum_cut networkx.algorithms.flow.dinitz_alg.dinitz networkx.algorithms.flow.maxflow.minimum_cut_value networkx.algorithms.flow.gomory_hu.gomory_hu_tree networkx.algorithms.flow.maxflow.maximum_flow_value networkx.algorithms.flow.shortestaugmentingpath.shortest_augmenting_path

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#312
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