Optimal General Matchings

Given a graphG = (V,E) and for each vertex v ∈ V a subsetB(v) of the set {0, 1, . . . , dG(v)}, where dG(v) denotes the degree of vertex v in the graph G, a B-matching of G is any set F ⊆ E such that dF (v) ∈ B(v) for each vertex v, where dF (v) denotes the number of edges of F incident to v. The general matching problem asks the existence of aB-matching in a given graph. A setB(v) is said to have a gap of length p if there exists a natural number k ∈ B(v) such that k+1, . . . , k+p / ∈ B(v) and k+p+1 ∈ B(v). Without any restrictions the general matching problem is NP-complete. However, if no set B(v) contains a gap of length greater than 1, then the problem can be solved in polynomial time and Cornuejols [2] presented an algorithm for finding a B-matching, if it exists. In this paper we consider a version of the general matching problem, in which we are interested in finding a B-matching having a maximum (or minimum) number of edges. We present the first polynomial time algorithm for the maximum/minimumB-matching for the case when no set B(v) contains a gap of length greater than 1.