Deterministic algorithms for the Lovász Local Lemma

Karthekeyan Chandrasekaran, Navin Goyal, Bernhard Haeupler

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The Lovász Local Lemma [5] (LLL) is a powerful result in probability theory that states that the probability that none of a set of bad events happens is nonzero if the probability of each event is small compared to the number of events that depend on it. It is often used in combination with the probabilistic method for non-constructive existence proofs. A prominent application is to k-CNF formulas, where LLL implies that, if every clause in the formula shares variables with at most d ≤ 2k /e other clauses then such a formula has a satisfying assignment. Recently, a randomized algorithm to efficiently construct a satisfying assignment was given by Moser [13]. Subsequently Moser and Tardos [14] gave a randomized algorithm to construct the structures guaranteed by the LLL in a very general algorithmic framework. We address the main problem left open by Moser and Tardos of derandomizing these algorithms efficiently. Specifically, for a k-CNF formula with m clauses and d ≤ 2k/(1+∈) / e for some ∈ ∈ (0, 1), we give an algorithm that finds a satisfying assignment in time Õ(m2(1+1/∈))). This improves upon the deterministic algorithms of Moser and of Moser-Tardos with running times m Ω(k2) and mΩ(k·1/∈) which are superpolynomial for k = ω(1) and upon other previous algorithms which work only for d ≤ 2k/16/e. Our algorithm works efficiently for the asymmetric version of LLL under the algorithmic framework of Moser and Tardos [14] and is also parallelizable, i.e., has polylogarithmic running time using polynomially many processors.

Original languageEnglish (US)
Title of host publicationProceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms
Pages992-1004
Number of pages13
StatePublished - May 6 2010
Externally publishedYes
Event21st Annual ACM-SIAM Symposium on Discrete Algorithms - Austin, TX, United States
Duration: Jan 17 2010Jan 19 2010

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other21st Annual ACM-SIAM Symposium on Discrete Algorithms
CountryUnited States
CityAustin, TX
Period1/17/101/19/10

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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  • Cite this

    Chandrasekaran, K., Goyal, N., & Haeupler, B. (2010). Deterministic algorithms for the Lovász Local Lemma. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 992-1004). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).