On smoothed k-CNF formulas and the Walksat algorithm

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(2/k) 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. ∗University of Edinburgh . E-mail: [email protected]. †The Weizmann Institute. E-mail: [email protected]. Supported in part by The Israel Science Foundation (grant No. 873/08) ‡Carnegie Mellon University. E-mail: [email protected]. Supported in part by NSF grant DMS0753472 §Tel-Aviv University. E-mail: [email protected]. Research supported in part by a USA-Israel BSF Grant, by a grant from the Israel Science Foundation, and by Pazy Memorial Award. ¶Tel-Aviv University. E-mail: [email protected].