An Algorithmic Proof of the Lovasz Local Lemma via Resampling Oracles

TL;DR

This paper introduces a general algorithmic framework using resampling oracles to find outcomes avoiding certain events, extending the Lovasz Local Lemma to broader settings with practical applications.

Contribution

It presents a new efficient algorithm based on resampling oracles for the Lovasz Local Lemma in more general scenarios than previously possible.

Findings

Existence of resampling oracles in all scenarios where the Lovasz Local Lemma applies

Design of efficient resampling oracles for most known applications

New results for packings of Latin transversals, rainbow matchings, and rainbow spanning trees

Abstract

The Lovasz 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. Finding such an outcome by an efficient algorithm has been an active research topic for decades. Breakthrough work of Moser and Tardos (2009) presented an efficient algorithm for a general setting primarily characterized by a product structure on the probability space. In this work we present an efficient algorithm for a much more general setting. Our main assumption is that there exist certain functions, called resampling oracles, that can be invoked to address the undesired occurrence of the events. We show that, in all scenarios to which the original Lovasz Local Lemma applies, there exist resampling oracles, although they are not necessarily efficient. Nevertheless, for essentially all known applications of the Lovasz Local Lemma and its generalizations, we have designed efficient resampling oracles. As applications of these techniques, we present new results for packings of Latin transversals, rainbow matchings and rainbow spanning trees.