Maximum Matchings

Abstract

The maximum matching problem is a very well-studied combinatorial optimization problem. Given an undirected graph G=(V,E), a matching is a subset E′⊂E of the edge set such that no two edges in E′ share a common endpoint. The maximum matching problem asks for a matching of maximum cardinality. Such problems arise, e.g., in team planning when edges of a graph denote possible collaborations of workers and the aim is to find a biggest partition of the workers into teams of size 2. Therefore, matching problems have numerous generalizations to hypergraphs and weighted graphs, which will not be discussed in this chapter. The maximum matching problem should not be confused with the maximal matching problem, where the aim is to find a subset of edges which is maximum with respect to inclusion, i.e., no proper superset of the matching is a matching.