Optimal General Matchings

Given a graph \(G=(V,E)\) and for each vertex \(v \in V\) a subset B(v) of the set \(\{0,1,\ldots , d_G(v)\}\), where \(d_G(v)\) denotes the degree of vertex v in the graph G, a B-matching of G is any set \(F \subseteq E\) such that \(d_F(v) \in B(v)\) for each vertex v, where \(d_F(v)\) denotes the number of edges of F incident to v. The general matching problem asks the existence of a B-matching in a given graph. A set B(v) is said to have a gap of length p if there exists a number \(k \in B(v)\) such that \(k+1, \ldots , k+p \notin B(v)\) and \(k+p+1 \in B(v)\). 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 [5] 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.

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