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

Abstract

The Lovász Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection 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 epd < 1, then a configuration avoiding all 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.

Pliable Index Coding via Conflict-Free Colorings of Hypergraphs

“…The main difficulty in getting a deterministic algorithm is the presence of the Local Lemma (Lemma 6) in the proof. Derandomization of the Local Lemma to provide an constructive algorithm has been studied [17], [18]. Applying Theorem 1.1 (1) in [18], we get a conflict-free coloring of a hypergraph using O(tΓ 1+ǫ t log Γ) colors, where t and Γ are as defined in Theorem 1 and ǫ > 0 is a constant.…”

“…Derandomization of the Local Lemma to provide an constructive algorithm has been studied [17], [18]. Applying Theorem 1.1 (1) in [18], we get a conflict-free coloring of a hypergraph using O(tΓ 1+ǫ t log Γ) colors, where t and Γ are as defined in Theorem 1 and ǫ > 0 is a constant. This suffices to get a deterministic polynomial time coloring algorithm for the hypergraph G in the proof of Theorem 4 using O(log 2 Γ) colors.…”

Nonrepetitive graph colouring

“…• Several recent papers have presented generalised Moser-Tardos frameworks improving the original work in various ways [1,80,82,83,85]. Nonrepetitive graph colouring has been a key test case here.…”