3-SAT Faster and Simpler - Unique-SAT Bounds for PPSZ Hold in General

The PPSZ algorithm by Paturi, Pudlak, Saks, and Zane [J. ACM, 52 (2005), pp. 337--364] is the fastest known algorithm for Unique $k$-SAT, where the input formula does not have more than one satisfying assignment. For $k\geq 5$ the same bounds hold for general $k$-SAT. We show that this is also the case for k=3,4, using a slightly modified PPSZ algorithm. We do the analysis by defining a cost for satisfiable conjunctive normal form formulas, which we prove to decrease in each PPSZ step by a certain amount. This improves our previous best bounds with Moser and Scheder [Proceedings of STACS, 2011, pp. 237--248] for 3-SAT to $O(1.308^n)$ and for 4-SAT to $O(1.469^n)$. Furthermore, our analysis is much simpler than the existing analysis of PPSZ for general $k$-SAT.