Deterministic search for CNF satisfying assignments in almost polynomial time

TL;DR

This paper presents a nearly polynomial-time deterministic algorithm for finding satisfying assignments in CNF formulas with many solutions, improving over previous methods that relied on pseudorandom generators.

Contribution

It introduces a novel framework connecting deterministic search with approximate counting, enabling faster algorithms for CNF satisfiability with many solutions.

Findings

Runs in n^{~(log log n)^2} time for formulas with many solutions

Outperforms previous PRG-based enumeration methods

Framework may be applicable to other derandomization problems

Abstract

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 $\mathrm{poly}(n)$-clause CNF formula $F$ that has at least $\varepsilon 2^n$ satisfying assignments, runs in time \[ n^{\tilde{O}(\log\log n)^2} \] for $\varepsilon \ge 1/\mathrm{polylog}(n)$ 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 $\varepsilon$. Our approach is based on a new general framework relating deterministic search and deterministic approximate counting, which we believe may find further applications.