Deterministic algorithms for the Lovász local lemma: Simpler, more general, and more parallel

The Lovász local lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection ℬ$$ \mathcal{B} $$ of “bad” events which are mostly independent and have low probability. A seminal algorithm of Moser and Tardos (J. ACM, 2010, 57, 11) (...

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Bibliographic Details
Published in:Random structures & algorithms Vol. 63; no. 3; pp. 716 - 752
Main Author: Harris, David G.
Format: Journal Article
Language:English
Published: New York John Wiley & Sons, Inc 01.10.2023
Wiley Subscription Services, Inc
Subjects:
Algorithms
Coloring
derandomization
independent transversal
LLL
Lovász Local Lemma
Probability theory
strong coloring
ISSN:1042-9832, 1098-2418
Online Access:Get full text
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Summary:The Lovász local lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection ℬ$$ \mathcal{B} $$ of “bad” events which are mostly independent and have low probability. A seminal algorithm of Moser and Tardos (J. ACM, 2010, 57, 11) (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 (Combinatorica, 1985, 5, 241–245); this is more powerful than alternate LLL criteria and also leads to cleaner and more legible bounds. Second, we provide parallel algorithms with much greater flexibility. 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.
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.21152