Approximating the permanent is in RNC

Abstract

In this work we look into the parallelization (in the NC sense) of the Markov Chain approach to almost random generation, as described by Sinclair (1993). We give a RNC almost uniform generator for perfect matchings in dense bipartite graphs, and its RNC reduction to its counting problem. As a consequence we can classify the problem of getting an approximation scheme to the permanent of a boolean matrix in RNC. We also state other problems for which the same technique applies.