On smoothed k-CNF formulas and the Walksat algorithm

Amin Coja-Oghlan, Uriel Feige, Alan Frieze, Michael Krivelevich, Dan Vilenchik

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

17 Scopus citations


In this paper we study the model of ε-smoothed k-CNF formulas. Starting from an arbitrary instance F with n variables and m = dn clauses, apply the ε-smoothing operation of flipping the polarity of every literal in every clause independently at random with probability ε. Keeping ε and k fixed, and letting the density d = m/n grow, it is rather easy to see that for d ≥ ε-k ln 2, F becomes whp unsatisfiable after smoothing. We show that a lower density that behaves roughly like ε-k+1 suffices for this purpose. We also show that our bound on d is nearly best possible in the sense that there are k-CNF formulas F of slightly lower density that whp remain satisfiable after smoothing. One consequence of our proof is a new lower bound of Ω(2k/k 2) on the density up to which Walksat solves random k-CNFs in polynomial time whp. We are not aware of any previous rigorous analysis showing that Walksat is successful at densities that are increasing as a function of k.

Original languageEnglish
Title of host publicationProceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Print)9780898716801
StatePublished - 1 Jan 2009
Externally publishedYes
Event20th Annual ACM-SIAM Symposium on Discrete Algorithms - New York, NY, United States
Duration: 4 Jan 20096 Jan 2009

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference20th Annual ACM-SIAM Symposium on Discrete Algorithms
Country/TerritoryUnited States
CityNew York, NY

ASJC Scopus subject areas

  • Software
  • Mathematics (all)


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