PPSZ is better than you think

PPSZ, for long time the fastest known algorithm for k-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 k ≥ 3. For Unique-3-SAT we bound its running time by O(1.306973n), which is somewhat better than the algorithm of Hansen, Kaplan, Zamir, and Zwick, which runs in time O(1.306995n). Before that, the best known upper bound for Unique-3-SAT was O(1.3070319n). 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.