A Parallel Algorithm for Sampling Matchings from an Almost Uniform Distribution

In this paper we present a randomized parallel algorithm to sample matchings from an almost uniform distribution on the set of matchings of all sizes in a graph. First we prove that the direct NC simulation of the sequential Markov chain technique for this problem is P-complete. Afterwards we present a randomized parallel algorithm for the problem. The technique used is based on the definition of a genetic system that converges to the uniform distribution. The system evolves according to a non-linear equation. Little is known about the convergence of these systems. We can define a non-linear system which converges to a stationary distribution under quite natural conditions. We prove convergence for the system corresponding to the almost uniform sampling of matchings in a graph (up to know the only known convergence for non-linear systems for matchings was matchings on a tree [RSW92]). We give empirical evidence that the system converges faster, in polylogarithmic parallel time.