Chain, Generalization of Covering Code, and Deterministic Algorithm for k-SAT

We present the current fastest deterministic algorithm for k-SAT, improving the upper bound (2-2/k)^{n + o(n)} due to Moser and Scheder in STOC 2011. The algorithm combines a branching algorithm with the derandomized local search, whose analysis relies on a special sequence of clauses called chain, and a generalization of covering code based on linear programming. We also provide a more intelligent branching algorithm for 3-SAT to establish the upper bound 1.32793^n, improved from 1.3303^n

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