An Algorithmic Proof of the Lovász Local Lemma via Resampling Oracles

Abstract

The Lovász 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. The breakthrough work of Moser [A constructive proof of the Lovász local lemma, in Proceedings of the ACM International Symposium on Theory of Computing, 2009, pp. 343--350] and Moser and Tardos [J. ACM, 57 (2010), 11] 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 Lovász local lemma applies, there exist resampling oracles, although they are not necessarily efficient. Nevertheless, for essentially all known applications of the Lovász local lemma and its generalizations, we have designed efficient resampling oracles. As an application of these techniques, we present a new result on packings of rainbow spanning trees.