Sequential importance sampling for estimating expectations over the space of perfect matchings

This paper makes three contributions to estimating the number of per-fect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a (1 +/- e)-approximation for the number of perfect matchings of a A.-dense bipartite graph, using O (n 1-2A. A. E-2 ) samples. With size n on each side and for 21 > A. > 0, a A.-dense bipartite graph has all degrees greater than (A. + 21)n. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements. Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.