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PPSZ for k ≥ 5: More Is Better

Published:12 September 2019Publication History
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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)| = 2δ n solutions, then PPSZ finds a satisfying assignment with probability at least 2ckno(n) + f(δ) n. Here, 2ckno(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 BB. 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.

References

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    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 11, Issue 4
      December 2019
      252 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/3331049
      Issue’s Table of Contents

      Copyright © 2019 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 12 September 2019
      • Accepted: 1 May 2019
      • Revised: 1 April 2019
      • Received: 1 July 2018
      Published in toct Volume 11, Issue 4

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