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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 n solutions, then PPSZ finds a satisfying assignment with probability at least 2ck n o(n) + f() n. Here, 2ck 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| / 2n 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 (minS: |S| = s maxB 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.
ACM Transactions on Computation Theory (TOCT) – Association for Computing Machinery
Published: Sep 12, 2019
Keywords: Boolean satisfiability
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