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 $(2-2\mathrm{/}k{)}^{n+o(n)}$ dues to Moser and Scheder [STOC’11]. 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 ingenious branching algorithm for $3$-SAT to establish the upper bound ${1.32793}^n$, improved from ${1.3303}^n$.