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 ε2ⁿ satisfying assignments, runs in time n(Õ(log log n)²) for ε ≥ 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 [DETT10], this takes time n^{Ω̃(log n)} even for constant ε. Our approach is based on a new general framework relating deterministic search and deterministic approximate counting, which we believe may find further applications.