An Improvement of the Lov\'asz Local Lemma via Cluster Expansion

TL;DR

This paper improves the Lovász Local Lemma by leveraging connections with the independent set polynomial and polymer gas partition functions, leading to tighter bounds in combinatorial problems like Latin transversals and k-SAT.

Contribution

It introduces an enhanced version of the Lovász Local Lemma using cluster expansion techniques and recent results on partition function analyticity.

Findings

Tighter bounds for Latin transversals

Improved conditions for k-SAT satisfiability

Enhanced probabilistic method applications

Abstract

An old result by Shearer relates the Lov\'asz Local Lemma with the independent set polynomial on graphs, and consequently, as observed by Scott and Sokal, with the partition function of the hard core lattice gas on graphs. We use this connection and a recent result on the analyticity of the logarithm of the partition function of the abstract polymer gas to get an improved version of the Lov\'asz Local Lemma. As applications we obtain tighter bounds on conditions for the existence of latin transversal matrices and the satisfiability of k-SAT forms.