Beyond the Lovasz Local Lemma: Point to Set Correlations and Their Algorithmic Applications

TL;DR

This paper introduces a unified convergence condition based on point-to-set correlations that analyzes resampling, backtracking, and hybrid algorithms for constraint satisfaction problems, simplifying analysis and extending algorithmic applications.

Contribution

It presents a novel correlation-based convergence condition that unifies and extends analysis of various local search algorithms, including hybrid methods, in the context of the Lovasz Local Lemma.

Findings

Unified analysis of resampling, backtracking, and hybrid algorithms.

Simplified entropy compression analysis.

Extended vertex coloring algorithms to broader graph classes.

Abstract

Following the groundbreaking algorithm of Moser and Tardos for the Lovasz Local Lemma (LLL), there has been a plethora of results analyzing local search algorithms for various constraint satisfaction problems. The algorithms considered fall into two broad categories: resampling algorithms, analyzed via different algorithmic LLL conditions; and backtracking algorithms, analyzed via entropy compression arguments. This paper introduces a new convergence condition that seamlessly handles resampling, backtracking, and hybrid algorithms, i.e., algorithms that perform both resampling and backtracking steps. Unlike all past LLL work, our condition replaces the notion of a dependency or causality graph by quantifying point-to-set correlations between bad events. As a result, our condition simultaneously: (i)~captures the most general algorithmic LLL condition known as a special case; (ii)~significantly simplifies the analysis of entropy compression applications; (iii)~relates backtracking algorithms, which are conceptually very different from resampling algorithms, to the LLL; and most importantly (iv)~allows for the analysis of hybrid algorithms, which were outside the scope of previous techniques. We give several applications of our condition, including a new hybrid vertex coloring algorithm that extends the recent breakthrough result of Molloy for coloring triangle-free graphs to arbitrary graphs.