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

TL;DR

This paper demonstrates that a sequential importance sampling algorithm can efficiently approximate the number of perfect matchings in dense bipartite graphs, with practical applications in counting Latin squares, card games, and stochastic block models.

Contribution

It proves polynomial-time approximation for dense bipartite graphs and introduces a preprocessing step to improve practical performance, along with three real-world applications.

Findings

Algorithm provides a $(1\u00b1\u03b5)$-approximation with polynomial samples.

Preprocessing significantly reduces computational effort.

Successfully applied to Latin squares, card games, and stochastic block models.

Abstract

This paper makes three contributions to estimating the number of perfect 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 $(1\pm\epsilon)$-approximation for the number of perfect matchings of a $\lambda$-dense bipartite graph, using $O(n^{\frac{1-2\lambda}{\lambda}\epsilon^{-2}})$ samples. With size $n$ on each side and for $\frac{1}{2}>\lambda>0$, a $\lambda$-dense bipartite graph has all degrees greater than $(\lambda+\frac{1}{2})n$. Second, practical applications of the algorithm require 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.