A Simpler Exponential-Time Approximation Algorithm for MAX-k-SAT

TL;DR

This paper introduces a simple exponential-time approximation algorithm for MAX-k-SAT that improves speed over previous methods by leveraging random sampling and probabilistic analysis.

Contribution

The paper presents a straightforward polynomial-space exponential-time algorithm for MAX-k-SAT that is slightly faster than prior approximation algorithms.

Findings

The algorithm achieves a (1-ε)-approximation for MAX-k-SAT.

Random sampling finds near-optimal assignments efficiently.

Exponential number of assignments satisfy near-optimal clause fractions.

Abstract

We present an extremely simple polynomial-space exponential-time $(1-\varepsilon)$-approximation algorithm for MAX-k-SAT that is (slightly) faster than the previous known polynomial-space $(1-\varepsilon)$-approximation algorithms by Hirsch (Discrete Applied Mathematics, 2003) and Escoffier, Paschos and Tourniaire (Theoretical Computer Science, 2014). Our algorithm repeatedly samples an assignment uniformly at random until finding an assignment that satisfies a large enough fraction of clauses. Surprisingly, we can show the efficiency of this simpler approach by proving that in any instance of MAX-k-SAT (or more generally any instance of MAXCSP), an exponential number of assignments satisfy a fraction of clauses close to the optimal value.