Deterministic Search for CNF Satisfying Assignments in Almost Polynomial Time

We consider the fundamental derandomization problem of deterministically finding a satisfying assignment to a CNF formula that has many satisfying assignments. We give a deterministic algorithm which, given an n-variable \poly(n)-clause CNF formula F that has at least ≥ 2^n satisfying assignments, runs in time \[ n^{\tilde{O}(\log\log n)^2} \] for ≥ \ge 1/\polylog(n) and outputs a satisfying assignment of F. Prior to our work the fastest known algorithm for this problem was simply to enumerate over all seeds of a pseudorandom generator for CNFs; using the best known PRGs for CNFs \cite{DETT10, this takes time n^{\tilde{Ω}(\log n)} even for constant ≥. Our approach is based on a new general framework relating deterministic search and deterministic approximate counting, which we believe may find further applications.