The Cardinality Matching Problem in Bipartite Graphs

Abstract

Given a graph G = (V, E) a 1-matching (or simply matching) in G is a subset M ⊆ E such that each vertex v ∈ V is incident with at most one edge from M . By M we denote the set of all matchings in G, where M ≠ Ø since Ø ∈ M holds. By |M| we denote the cardinality of M . Then the maximum cardinality matching problem (CMP) is to find a matching in G the cardinality of which is as large as possible i.e. $$ \max \;\{ |M|\;|M \in M\} . $$ (1) An optimal solution M of CMP is called a maximum cardinality matching. A matching M ∈ M is called perfect iff every node v 2208 V is incident to exactly one edge from M . If G = (Vs, Vi, E) is a bipartite graph then a perfect matching is also called an assignment. Obviously any perfect matching solves CMP. In general we will denote by v(G) the matching number of G i.e. the cardinality of a maximum cardinality matching.