Let G = (U,V,E) be a bipartite graph with \U\=\V\= n. The factor size of G, f is the maximum number of edge disjoint perfect matchings in G. We characterize the complexity of counting the number of perfect matchings in classes of graphs parameterized by factor size. We describe the simple algorithm, which is an approximation algorithm for the permanent that is a natural simplification of the algorithm suggested by Broder (1986) and analyzed by Jerrum and Sinclair (1988a, b). Compared to the algorithm by Jerrum and Sinclair (1988a, b), the simple algorithm achieves a polynomial speed up in the running time to compute the permanent. A combinatorial lemma is used to prove that the simple algorithm runs in time n(O(n/f)). Thus: (1) for all constants alpha > 0, the simple algorithm runs in polynomial time for graphs with factor size at least alpha-n; (2) for some constant c, the simple algorithm is the fastest known approximation for graphs with factor size at least c log n. (Compare with the approximation algorithms described in Karmarkar et al. (1988).) We prove the following complementary hardness results. For functions f such that 3 less-than-or-equal-to f(n) less-than-or-equal-to n-3, the exact counting problem for f (n)-regular bipartite graphs is #P-complete. For any epsilon>0, for any function f such that 3 less-than-or-equal-to f(n) less-than-or-equal-to n1-epsilon, approximate counting for f(n)-regular bipartite graphs is as hard as approximate counting for all bipartite graphs. An announcement of these results appears in Dagum et al. (1988).
