Maximum weight matching

In computer science and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized.

Maximum weight matching of 2 graphs. The first is also a perfect matching, while the second is far from it with 4 vertices unaccounted for, but has high value weights compared to the other edges in the graph.

A special case of the maximum weight matching problem is the assignment problem, in which the graph is a bipartite graph and the matching must have cardinality equal to that of the smaller of the two partitions. Another special case is the problem of finding a maximum cardinality matching on an unweighted graph: this corresponds to the case where all edge weights are the same.

Algorithms

There is a ${\displaystyle O(V^{2}E)}$ time algorithm to find a maximum matching or a maximum weight matching in a graph that is not bipartite; it is due to Jack Edmonds, is called the paths, trees, and flowers method or simply Edmonds' algorithm, and uses bidirected edges. A generalization of the same technique can also be used to find maximum independent sets in claw-free graphs.

More elaborate algorithms exist and are reviewed by Duan and Pettie^[1] (see Table III). Their work proposes an approximation algorithm for the maximum weight matching problem, which runs in linear time for any fixed error bound.