Approximating the Permanent

Abstract

A randomised approximation scheme for the permanent of a 0–1s presented. The task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph; the latter is accomplished by simulating a Markov chain whose states are the matchings in the graph. For a wide class of 0–1 matrices the approximation scheme is fully-polynomial, i.e., runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output. This class includes all dense matrices (those that contain sufficiently many 1’s) and almost all sparse matrices in some reasonable probabilistic model for 0–1 matrices of given density. For the approach sketched above to be computationally efficient, the Markov chain must be rapidly mixing: informally, it must converge in a short time to its stationary distribution. A major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely complex structure. The techniques used seem to have general applicability, and are applied again in the paper to validate a fully-polynomial randomised approximation scheme for the partition function of an arbitrary monomer-dimer system.