A polynomial algorithm for b-matchings: An alternative approach

Abstract

A b-matching of a given graph G is an assignment of integer weights to the edges of G so that the sum of the weights on the edges incident with a vertex v is at most b v (b denotes the vectors of b v's). When b v = 1 for all vertices v in G, then b-matchings are the usual matchings. The b-matching problem asks for a b-matching of maximum cost where the edges of G have been assigned costs and the cost of a b-matching is the sum of the weights times the costs. We do not assume G to be bipartite. We present a polynomial algorithm for the b-matching problem differing from the algorithm of Cunningham and Marsh (1979). In fact, the new algorithm is strongly polynomial using the new strongly polynomial minimum cost flow algorithm of Tardos (we actually use the algorithm of Orlin (1984)) to obtain a half-integral optimal solution. An integral solution (close to being optimal) is obtained from it which is used as input to Pulleyblank's (1973) b-matching algorithm.