Approximately counting independent sets in bipartite graphs via graph containers

TL;DR

This paper introduces new algorithms based on graph container methods to efficiently approximate the number of independent sets in bipartite graphs, extending previous results to broader classes of graphs and weights.

Contribution

It develops algorithmic versions of Sapozhenko's graph container methods, providing FPTAS for counting independent sets in various bipartite graph classes under weaker expansion conditions.

Findings

FPTAS for independent sets in $d$-regular bipartite graphs with weak expansion

Extension to weighted independent sets and the hard-core model

Approximation algorithm for all $d$-regular bipartite graphs with subexponential runtime

Abstract

By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to $d$-regular, bipartite graphs satisfying a weak expansion condition: when $d$ is constant, and the graph is a bipartite $\Omega( \log^2 d/d)$-expander, we obtain an FPTAS for the number of independent sets. Previously such a result for $d>5$ was known only for graphs satisfying the much stronger expansion conditions of random bipartite graphs. The algorithm also applies to weighted independent sets: for a $d$-regular, bipartite $\alpha$-expander, with $\alpha>0$ fixed, we give an FPTAS for the hard-core model partition function at fugacity $\lambda=\Omega(\log d / d^{1/4})$. Finally we present an algorithm that applies to all $d$-regular, bipartite graphs, runs in time $\exp\left( O\left( n \cdot \frac{ \log^3 d }{d } \right) \right)$, and outputs a $(1 + o(1))$-approximation to the number of independent sets.