Deterministic algorithms for the Lovasz Local Lemma: simpler, more general, and more parallel

TL;DR

This paper introduces simplified, more general, and parallel deterministic algorithms for the Lovász Local Lemma, improving upon prior methods by applying to Shearer's criterion and offering greater flexibility and derandomization.

Contribution

The authors develop deterministic algorithms for the Lovász Local Lemma that are more powerful, flexible, and parallelizable, extending applicability to Shearer's criterion and derandomizing the MT-distribution.

Findings

Algorithms match the efficiency of previous randomized methods.

Applicable to a wider range of bad-event structures.

Enable deterministic solutions for problems like non-repetitive coloring.

Abstract

The Lov\'{a}sz Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection $\mathcal B$ of "bad" events which are mostly independent and have low probability. In its simplest "symmetric" form, it asserts that whenever a bad-event has probability $p$ and affects at most $d$ bad-events, and $e p d < 1$, then a configuration avoiding all $\mathcal B$ exists. A seminal algorithm of Moser & Tardos (2010) gives nearly-automatic randomized algorithms for most constructions based on the LLL. However, deterministic algorithms have lagged behind. We address three specific shortcomings of the prior deterministic algorithms. First, our algorithm applies to the LLL criterion of Shearer (1985); this is more powerful than alternate LLL criteria and also removes a number of nuisance parameters and leads to cleaner and more legible bounds. Second, we provide parallel algorithms with much greater flexibility in the functional form of of the bad-events. Third, we provide a derandomized version of the MT-distribution, that is, the distribution of the variables at the termination of the MT algorithm. We show applications to non-repetitive vertex coloring, independent transversals, strong coloring, and other problems. These give deterministic algorithms which essentially match the best previous randomized sequential and parallel algorithms.