max_weight_matching

max_weight_matching(G, maxcardinality=False, weight='weight')

Compute a maximum-weighted matching of G.

A matching is a subset of edges in which no node occurs more than once. The weight of a matching is the sum of the weights of its edges. A maximal matching cannot add more edges and still be a matching. The cardinality of a matching is the number of matched edges.

Parameters:

GNetworkX graph

Undirected graph

maxcardinality: bool, optional (default=False)

If maxcardinality is True, compute the maximum-cardinality matching with maximum weight among all maximum-cardinality matchings.

weight: string, optional (default=’weight’)

Edge data key corresponding to the edge weight. If key not found, uses 1 as weight.

Returns:

matchingset

A maximal matching of the graph.

Notes

If G has edges with weight attributes the edge data are used as weight values else the weights are assumed to be 1.

This function takes time O(number_of_nodes ** 3).

If all edge weights are integers, the algorithm uses only integer computations. If floating point weights are used, the algorithm could return a slightly suboptimal matching due to numeric precision errors.

This method is based on the “blossom” method for finding augmenting paths and the “primal-dual” method for finding a matching of maximum weight, both methods invented by Jack Edmonds [1].

Bipartite graphs can also be matched using the functions present in networkx.algorithms.bipartite.matching.

References

[1]

“Efficient Algorithms for Finding Maximum Matching in Graphs”, Zvi Galil, ACM Computing Surveys, 1986.

Examples

>>> G = nx.Graph()
>>> edges = [(1, 2, 6), (1, 3, 2), (2, 3, 1), (2, 4, 7), (3, 5, 9), (4, 5, 3)]
>>> G.add_weighted_edges_from(edges)
>>> sorted(nx.max_weight_matching(G))
[(2, 4), (5, 3)]