The polynomially bounded perfect matching problem Is in NC²

Abstract

The perfect matching problem is known to be in P, in randomized NC, and it is hard for NL. Whether the perfect matching problem is in NC is one of the most prominent open questions in complexity theory regarding parallel computations. Grigoriev and Karpinski [GK87] studied the perfect matching problem for bipartite graphs with polynomially bounded permanent. They showed that for such bipartite graphs the problem of deciding the existence of a perfect matchings is in NC², and counting and enumerating all perfect matchings is in NC³. For general graphs with a polynomially bounded number of perfect matchings, they show both problems to be in NC³. In this paper we extend and improve these results. We show that for any graph that has a polynomially bounded number of perfect matchings, we can construct all perfect matchings in NC². We extend the result to weighted graphs.