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

Abstract

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Deterministic algorithms for the Lovász Local Lemma: simpler, more general, and more parallelDavid G. HarrisDavid G. Harrispp.1744 - 1779Chapter DOI:https://doi.org/10.1137/1.9781611977073.71PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract 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) (which we call the MT algorithm) 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 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. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771