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On smoothed k-CNF formulas and the Walksat algorithm

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

Abstract

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/k2) 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

Topics: QA
Publisher: SIAM
Year: 2009
DOI identifier: 10.1137/1.9781611973068.50
OAI identifier: oai:wrap.warwick.ac.uk:43236
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