The Local Lemma is tight for SAT

H. Gebauer, T. Szabó, G. Tardos

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

18 Citations (Scopus)

Abstract

We construct unsatisfiable k-CNF formulas where every clause has k distinct literals and every variable appears in at most (2/e + o(l))2k/k clauses. The lopsided Local Lemma shows that our result is asymptotically best possible: every k-CNF formula where every variable appears in at most 2 k+1/e(k+1) - 1 clauses is satisfiable. The determination of this extremal function is particularly important as it represents the value where the k-SAT problem exhibits its complexity hardness jump: from having every instance being a YES-instance it becomes NP-hard just by allowing each variable to occur in one more clause. The asymptotics of other related extremal functions are also determined. Let l(k) denote the maximum number, such that every k-CNF formula with each clause containing k distinct literals and each clause having a common variable with at most l(k) other clauses, is satisfiable. We establish that the bound on l(k) obtained from the Local Lemma is asymptotically optimal, i.e., l(k) = ( 1/e +o(1))2k The constructed formulas are all in the class MU(1) of minimal unsatisfiable formulas having one more clause than variables and thus they resolve these asymptotic questions within that class as well. The SAT-formulas are constructed via the binary trees of [10]. In order to construct the trees a continuous setting of the problem is defined, giving rise to a differential equation. The solution of the equation diverges at 0, which in turn implies that the binary tree obtained from the discretization of this solution has the required properties.

Original languageEnglish
Title of host publicationProceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011
Pages664-674
Number of pages11
Publication statusPublished - May 12 2011
Event22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011 - San Francisco, CA, United States
Duration: Jan 23 2011Jan 25 2011

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011
CountryUnited States
CitySan Francisco, CA
Period1/23/111/25/11

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ASJC Scopus subject areas

  • Software
  • Mathematics(all)

Cite this

Gebauer, H., Szabó, T., & Tardos, G. (2011). The Local Lemma is tight for SAT. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011 (pp. 664-674). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).