Randomized sequential importance sampling for estimating the number of perfect matchings in bipartite graphs

TL;DR

This work introduces randomized sequential importance sampling methods to estimate the number of perfect matchings in bipartite graphs, and develops central limit theorems for the log-likelihood ratio under several sampling orders. By analyzing Fibonacci, 2-Fibonacci, and distance-2 matchings, it derives explicit mean and variance growth rates and demonstrates how KL-based sample-size criteria $N^*$ can drastically reduce the number of samples needed compared to classical variance-based criteria. The results include CLTs for random, fixed, and greedy order algorithms, the construction of almost-perfect variants with constant variance, and quantitative comparisons with Markov chain Monte Carlo approaches in practical regimes. Collectively, the paper provides a framework for efficient, theory-backed approximation of #P-complete counting problems in structured bipartite graphs. The demonstrated concentration phenomena and exact asymptotics offer guidance for applying importance sampling to related counting and sampling tasks, such as contingency tables and degree-sequence graphs.

Abstract

We introduce and study randomized sequential importance sampling algorithms for estimating the number of perfect matchings in bipartite graphs. In analyzing their performance, we establish various non-standard central limit theorems. We expect our methods to be useful for other applied problems.

Figures (8)

• Figure 1: A bipartite graph and its three perfect matchings.
• Figure 2: The 5 Fibonacci matchings of size $n=4$.
• Figure 3: Perfect matchings in this graph (see \ref{['F_fib_n4']}) correspond to Fibonacci permutations in ${\mathcal{F}}_{4,1}$.
• Figure 4: $Y(z)$ has singularities at $z_r\doteq0.3720$ and $z'\doteq-1.0079$, where $e^{(1+z)^2}(2/3e-\int_{1}^{1+z}e^{-u^2}du)=1$.
• Figure 5: Perfect matchings in this graph correspond to $2$-Fibonacci permutations in ${\mathcal{F}}_{5,2}$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 2.1: CD18 Theorem 1.1
• Lemma 3.1: NR04 Corollary 5.2
• Theorem 3.2
• Lemma 3.3
• proof : Proof of \ref{['T_Ar']}
• proof : Proof of \ref{['L_Ar']}
• Theorem 3.4
• proof : Proof of \ref{['T_Af']}
• Theorem 3.5