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

TL;DR

The paper tackles the problem of approximating MAX-k-SAT with a simple, polynomial-space randomized algorithm. By proving a general lower bound on the number of near-optimal assignments for MAXCSP (via an entropy-based argument), the authors justify a straightforward sampling strategy: repeatedly sample a random assignment, evaluate its clause satisfaction, and output the best found. They derive an explicit exponential-time bound for MAX-k-SAT that improves upon the Hirsch03 and related polynomial-space methods when k is fixed, showing the approach is both simpler and competitive in the exponential regime. The results extend to weighted MAXCSP, offering a clean, space-efficient framework with practical implications for approximation in CSPs and potential connections to broader algorithmic paradigms.

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.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• proof : Proof of \ref{['th:lb']}
• Theorem 3
• Corollary 1
• proof
• Corollary 2
• proof