PPSZ is better than you think

Abstract

PPSZ, for long time the fastest known algorithm for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> -SAT, works by going through the variables of the input formula in random order; each variable is then set randomly to 0 or 1, unless the correct value can be inferred by an efficiently implementable rule (like small-width resolution; or being implied by a small set of clauses). We show that PPSZ performs exponentially better than previously known, for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k\geq 3$</tex> . For Unique-3-SAT we bound its running time by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(1.306973^{n})$</tex> , which is somewhat better than the algorithm of Hansen, Kaplan, Zamir, and Zwick. All improvements are achieved without changing the original PPSZ. The core idea is to pretend that PPSZ does not process the variables in uniformly random order, but according to a carefully designed distribution. We write “pretend” since this can be done without any actual change to the algorithm. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> The full version of this paper can be found under https://eccc.weizmann.ac.il/report/2021/069/