maximum matching

A maximal matching is a matching M of a graph G with the property that if any edge not in M is added to M, it is no longer a matching, that is, M is maximal if it is not a subset of any other matching in graph G. We cover Blossom, Hungarian and Hopcroft Karp algorithm

The blossom algorithm, sometimes called the Edmonds' matching algorithm, can be used on any graph to construct a maximum matching. The blossom algorithm improves upon the Hungarian algorithm by shrinking cycles in the graph to reveal augmenting paths. The blossom algorithm will work on any graph.

The Hungarian maximum matching algorithm, also called the Kuhn-Munkres algorithm, is a O(V3) algorithm that can be used to find maximum-weight matchings in bipartite graphs, which is sometimes called the assignment problem. A bipartite graph can easily be represented by an adjacency matrix

The Hopcroft–Karp algorithm is an algorithm that takes as input a bipartite graph and produces as output a maximum cardinality matching, it runs in O(E√V) time in worst case.