Short proofs for generalizations of the Lov\'asz Local Lemma: Shearer's condition and cluster expansion

TL;DR

This paper provides concise proofs of advanced generalizations of the Lovász Local Lemma, specifically Shearer's and the cluster expansion lemmas, and improves bounds for the symmetric case.

Contribution

It introduces short proofs for Shearer's lemma and the cluster expansion lemma in their lopsided forms, and refines probability bounds in the symmetric local lemma.

Findings

Short proofs of Shearer's and cluster expansion lemmas

Improved probability bound of 1/ed in the symmetric local lemma

Extension of the local lemma to more general conditions

Abstract

The Lov\'asz Local Lemma is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events. Over the years, more general conditions have been discovered under which the conclusion of the lemma continues to hold. In this note we provide short proofs of two of those more general results: Shearer's lemma and the cluster expansion lemma, in their "lopsided" form. We conclude by using the cluster expansion lemma to prove that the symmetric form of the local lemma holds with probabilities bounded by $1/ed$, rather than the bound $1/e(d+1)$ required by the traditional proofs.