Theoretical Analysis of Sequential Importance Sampling Algorithms for a Class of Perfect Matching Problems

TL;DR

This paper provides a theoretical analysis of sequential importance sampling algorithms for counting perfect matchings in bipartite graphs, deriving bounds on sample complexity and analyzing permutation moments through advanced probabilistic methods.

Contribution

It introduces new theoretical bounds on the sample size needed for accurate estimation and employs novel techniques involving permutation moments and Markov chains.

Findings

Derived precise bounds on sample requirements for accuracy.

Computed moments of permutation statistics using generating functions.

Analyzed perfect matchings via time-inhomogeneous Markov chains.

Abstract

This paper analyzes the performance of sequential importance sampling algorithms for estimating the number of perfect matchings in bipartite graphs. Precise bounds on the number of samples required to yield an accurate estimate are derived. In doing so, moments of permutation statistics are computed using generating functions and nonstandard limit theorems are derived by expressing perfect matchings as a time-inhomogeneous Markov chain.