Approximate counting and sampling via local central limit theorems

TL;DR

This paper introduces a new framework leveraging local central limit theorems to develop efficient algorithms for approximately counting and sampling matchings and independent sets of certain sizes in bounded-degree graphs.

Contribution

It presents the first FPTAS and randomized sampling algorithms for counting and sampling matchings and independent sets below specific size thresholds, based on a novel local CLT approach.

Findings

FPTAS for matchings of size up to (1-δ)m^*(G)

FPTAS for independent sets of size up to (1-δ)α_c(Δ)n

Quasi-linear time randomized algorithms for sampling

Abstract

We give an FPTAS for computing the number of matchings of size $k$ in a graph $G$ of maximum degree $\Delta$ on $n$ vertices, for all $k \le (1-\delta)m^*(G)$, where $\delta>0$ is fixed and $m^*(G)$ is the matching number of $G$, and an FPTAS for the number of independent sets of size $k \le (1-\delta) \alpha_c(\Delta) n$, where $\alpha_c(\Delta)$ is the NP-hardness threshold for this problem. We also provide quasi-linear time randomized algorithms to approximately sample from the uniform distribution on matchings of size $k \leq (1-\delta)m^*(G)$ and independent sets of size $k \leq (1-\delta)\alpha_c(\Delta)n$. Our results are based on a new framework for exploiting local central limit theorems as an algorithmic tool. We use a combination of Fourier inversion, probabilistic estimates, and the deterministic approximation of partition functions at complex activities to extract approximations of the coefficients of the partition function. For our results for independent sets, we prove a new local central limit theorem for the hard-core model that applies to all fugacities below $\lambda_c(\Delta)$, the uniqueness threshold on the infinite $\Delta$-regular tree.