Deterministic search for CNF satisfying assignments in almost polynomial time

01/10/2018

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by   Rocco A. Servedio, et al.

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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 ε 2^n satisfying assignments, runs in time n^Õ( 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( 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.