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 |F^{-1}(1)| \geq \eps 2^n, runs in time

 

n^{\tilde{O}(\log\log n)^2}

 

for \eps \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 [DETT10], this takes time n^{\tilde{\Omega}(\log n)} even for constant \eps.

 

Joint work with Rocco Servedio.