PPSZ for k ≥ 5: More Is Better

Abstract

We show that for k ≥ 5, the PPSZ algorithm for k -SAT runs exponentially faster if there is an exponential number of satisfying assignments. More precisely, we show that for every k ≥ 5, there is a strictly increasing function f : [0,1] → R with f (0) = 0 that has the following property. If F is a k -CNF formula over n variables and |sat(F)| = 2 δ n solutions, then PPSZ finds a satisfying assignment with probability at least 2 − c k n − o ( n ) + f (δ) n . Here, 2 − c k n − o ( n ) is the success probability proved by Paturi et al. [11] for k -CNF formulas with a unique satisfying assignment. Our proof rests on a combinatorial lemma: given a set S ⊆ { 0,1} n , we can partition { 0,1} n into subcubes such that each subcube B contains exactly one element of S . Such a partition B induces a distribution on itself, via Pr [ B ] = |B| / 2 n for each B ∈ B . We are interested in partitions that induce a distribution of high entropy. We show that, in a certain sense, the worst case (min S : |S| = s max B H ( B )) is achieved if S is a Hamming ball. This lemma implies that every set S of exponential size allows a partition of linear entropy. This in turn leads to an exponential improvement of the success probability of PPSZ.