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

2025-10-20

AI Summary

This paper addresses the MAX-k-SAT problem by proposing a minimalist (1−ε)-approximation algorithm that runs in exponential time O((k/ε)·2^{(1−ε)n}) while using only polynomial space. The method relies on repeated random assignment sampling coupled with refined probabilistic analysis: leveraging the structural property that assignments near the optimum form a high-density region in the solution space, it suffices to sample O((k/ε)·2^{(1−ε)n}) assignments to obtain a (1−ε)-approximate solution with high probability. Compared to prior algorithms by Hirsch (2003) and Escoffier et al. (2010), our approach achieves a constant-factor improvement in theoretical runtime while retaining polynomial space complexity—without resorting to intricate branching rules or memoization structures. The key innovation lies in identifying and exploiting the uniform near-optimality distribution over the assignment space, thereby unifying simplicity, efficiency, and analytical tractability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Sampling/Simulation-based Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Develops faster approximation algorithm for MAX-k-SAT
• Uses random sampling to find satisfying assignments
• Proves many solutions exist near optimal values

Innovation

• Random sampling for MAX-k-SAT approximation
• Polynomial-space exponential-time algorithm
• Uniform assignment sampling for clause satisfaction