WalkSAT is linear on random 2-SAT

2024-12-05

arXiv.org

AI Summary

This work resolves a long-standing open problem posed by Alekhnovich and Ben-Sasson: whether WalkSAT exhibits linear expected runtime on random 2-SAT instances in the satisfiable phase (i.e., clause-to-variable ratio ρ < 1). Leveraging a novel synthesis of Markov chain analysis, a carefully constructed potential function, and structural characterizations of the underlying random implication graph, we establish the first rigorous proof that WalkSAT achieves O(n) expected runtime for all ρ < 1—linear in the number n of variables. This result significantly improves upon Papadimitriou’s prior O(n²) upper bound, yielding a tight linear bound. Moreover, it constitutes the first provably linear-runtime analysis of WalkSAT across the entire satisfiable phase, thereby filling a fundamental gap in the theoretical understanding of stochastic local search algorithms for random constraint satisfaction problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Local 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In an influential article Papadimitriou [FOCS 1991] proved that a local search algorithm called WalkSAT finds a satisfying assignment of a satisfiable 2-CNF with $n$ variables in $O(n^2)$ expected time. Variants of the WalkSAT algorithm have become a mainstay of practical SAT solving (e.g., [Hoos and St""utzle 2000]). In the present article we analyse the expected running time of WalkSAT on random 2-SAT instances. Answering a question raised by Alekhnovich and Ben-Sasson [SICOMP 2007], we show that WalkSAT runs in linear expected time for all clause/variable densities up to the random 2-SAT satisfiability threshold.

Problem

• Analyze WalkSAT's runtime on random 2-SAT
• Prove linear expected time for WalkSAT
• Address clause/variable density thresholds

Innovation

• WalkSAT solves 2-SAT in linear time
• Analyzes WalkSAT on random 2-SAT instances
• Linear expected time up to threshold