Improving PPSZ for 3-SAT using Critical Variables

TL;DR

This paper enhances the PPSZ algorithm for 3-SAT by leveraging critical variables, achieving the fastest known exponential time bounds through novel combination of techniques.

Contribution

It introduces a new approach based on critical variables that improves the running time of PPSZ for 3-SAT and combines with recent methods for even faster algorithms.

Findings

Improved 3-SAT running time to O(1.32065^n)

Enhanced 4-SAT bound to O(1.46928^n)

Combined techniques outperform previous algorithms

Abstract

A critical variable of a satisfiable CNF formula is a variable that has the same value in all satisfying assignments. Using a simple case distinction on the fraction of critical variables of a CNF formula, we improve the running time for 3-SAT from O(1.32216^n) by Rolf [2006] to O(1.32153^n). Using a different approach, Iwama et al. [2010] very recently achieved a running time of O(1.32113^n). Our method nicely combines with theirs, yielding the currently fastest known algorithm with running time O(1.32065^n). We also improve the bound for 4-SAT from O(1.47390^n) [Iwama, Tamaki 2004] to O(1.46928^n), where O(1.46981^n) can be obtained using the methods of [Iwama, Tamaki 2004] and [Rolf 2006].