The Lovász Local Lemma and Satisfiability

  • Heidi Gebauer
  • Robin A. Moser
  • Dominik Scheder
  • Emo Welzl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5760)


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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Heidi Gebauer
    • 1
  • Robin A. Moser
    • 1
  • Dominik Scheder
    • 1
  • Emo Welzl
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland

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