On smoothed k-CNF formulas and the Walksat algorithm

In this paper we study the model of epsilon-smoothed k-CNF formulas. Starting from an arbitrary instance F with n variables and m = dn clauses, apply the epsilon-smoothing operation of flipping the polarity of every literal in every clause independently at random with probability epsilon. Keeping epsilon and k fixed, and letting the density d = m/n grow, it is rather easy to see that for d >= epsilon(-k) In 2, F becomes whp unsatisfiable after smoothing. We show that a lower density that behaves roughly like epsilon(-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 Omega(2(k)/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.