Approximating the Permanent of Graphs with Large Factors

Let G = (U; V; E) be a bipartite graph with jUj = jV j = 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 match-ings 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 simpliication of the algorithm suggested in Broder 86] and analyzed in Jerrum, Sinclair 88a, 88b]. Compared to the algorithm in Jerrum, Sinclair 88a, 88b], 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 > 0, 1 the simple algorithm runs in polynomial time for graphs with factor size at least 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 88].) We prove the following complementary hardness results. For functions f such that 3 f(n) n ? 3, the exact counting problem for f(n)-regular bipartite graphs is ]P-complete. For any > 0, for any function f such that 3 f(n) n 1? , 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 88].