Chapter

Efficient Algorithms

Volume 5760 of the series Lecture Notes in Computer Science pp 30-54

The Lovász Local Lemma and Satisfiability

  • Heidi GebauerAffiliated withInstitute of Theoretical Computer Science, ETH Zürich
  • , Robin A. MoserAffiliated withInstitute of Theoretical Computer Science, ETH Zürich
  • , Dominik SchederAffiliated withInstitute of Theoretical Computer Science, ETH Zürich
  • , Emo WelzlAffiliated withInstitute of Theoretical Computer Science, ETH Zürich

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Abstract

We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovász Local Lemma, a probabilistic lemma with many applications in combinatorics.

In positive terms, we are interested in simple combinatorial conditions which guarantee for a CNF formula to be satisfiable. The criteria obtained are nontrivial in the sense that even though they are easy to check, it is by far not obvious how to compute a satisfying assignment efficiently in case the conditions are fulfilled; until recently, it was not known how to do so. It is also remarkable that while deciding satisfiability is trivial for formulas that satisfy the conditions, a slightest relaxation of the conditions leads us into the territory of NP-completeness.

Several open problems remain, some of which we mention in the concluding section.