Approximating the permanent

Reviewer: David J. Aldous

In previous work [1], the authors described an interesting and important technique for “approximate counting” of large combinatorial sets. The ingredients are (1) construction of a reversible Markov chain on the set with uniform stationary distribution, (2) some proof of the “rapid mixing” property of the chain, and (3) a “self-reducibility” property of the sets in question. In this paper, they apply this technique to the problem of evaluating the permanent of a 0-1 valued matrix, which is equivalent to counting perfect matches in the associated adjacency graph. For a wide class of such matrices (including all sufficiently dense matrices and almost all matrices in the probabilistic sense), they show that a fully polynomial approximation scheme exists. The difficult part of the technique is proving the rapid mixing property, and this paper is perhaps the most complex example in which this proof has been successfully carried out.