The random $k$-SAT Gibbs uniqueness threshold revisited

2025-06-02

AI Summary

This work addresses the Gibbs uniqueness threshold for random $k$-SAT, aiming to rigorously validate the physical intuition that the replica-symmetric (RS) solution accurately characterizes the asymptotic number of satisfying assignments below this threshold. For $k geq 3$, the lower bound on this threshold has long remained imprecisely characterized. We combine probabilistic analysis, Gibbs measure construction on trees, coupling techniques, and the theory of local convergence of random graphs to establish, for the first time, that the RS formula asymptotically predicts the number of satisfying assignments throughout the Gibbs uniqueness regime. Moreover, we significantly improve the best-known lower bound on this threshold—especially for small $k$, where our bound substantially surpasses the seminal 2007 SODA result—yielding the tightest rigorous lower bound to date.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Graphical Models
• 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
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We prove that for any $kgeq3$ for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random $k$-SAT formulas is given by the `replica symmetric solution' predicted by physics methods [Monasson, Zecchina: Phys. Rev. Lett. (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any $kgeq3$, we derive new lower bounds on this threshold that improve over prior work [Montanari and Shah: SODA (2007)].The improvement is significant particularly for small $k$.

Problem

• Determine replica symmetric solution for random k-SAT
• Improve lower bounds on Gibbs uniqueness threshold
• Focus on small k cases for significant improvement

Innovation

• Uses replica symmetric solution
• Improves Gibbs uniqueness threshold bounds
• Focuses on small k values