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 $\mathcal B$ of "bad" events which are mostly independent and have low probability. In its simplest "symmetric" form, it asserts that whenever a badevent has probability $p$ and affects at most $d$ badevents, and $e p d < 1$, then a configuration avoiding all $\mathcal B$ exists. A seminal algorithm of Moser & Tardos (2010) gives nearlyautomatic 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 badevents. Third, we provide a derandomized version of the MTdistribution, that is, the distribution of the variables at the termination of the MT algorithm. We show applications to nonrepetitive vertex coloring, independent transversals, strong coloring, and other problems. These give deterministic algorithms which essentially match the best previous randomized sequential and parallel algorithms.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.08065
 Bibcode:
 2019arXiv190908065H
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 This superseded arxiv:1807.06672