Random local access for sampling k-SAT solutions

2024-09-06

arXiv.org

AI Summary

This work addresses approximate uniform sampling of satisfying assignments for a $k$-CNF formula $Phi$ under exponential clause density, targeting Random Local Access (RLA)—i.e., efficient marginal query for a single variable—without global storage of solutions. We propose the first sublinear-time RLA algorithm applicable to arbitrary $k$-SAT instances. It integrates local exploration, constraint propagation, and a Gibbs-distribution-based local simulation framework, augmented by an efficient rejection sampling strategy. Each variable query runs in $O(mathrm{polylog},n)$ time, and the output distribution is within total variation distance $varepsilon$ of the uniform distribution over satisfying assignments, without storing the entire solution space. This is the first successful extension of the RLA model to $k$-SAT—a canonical NP-hard sampling problem—surpassing prior limitations restricted to structured problems like graph coloring. Our result establishes that the solution space remains locally accessible even at high clause densities.

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

We present a sublinear time algorithm that gives random local access to the uniform distribution over satisfying assignments to an arbitrary k-CNF formula $Phi$, at exponential clause density. Our algorithm provides memory-less query access to variable assignments, such that the output variable assignments consistently emulate a single global satisfying assignment whose law is close to the uniform distribution over satisfying assignments to $Phi$. Such models were formally defined (for the more general task of locally sampling from exponentially sized sample spaces) in 2017 by Biswas, Rubinfeld, and Yodpinyanee, who studied the analogous problem for the uniform distribution over proper q-colorings. This model extends a long line of work over multiple decades that studies sublinear time algorithms for problems in theoretical computer science. Random local access and related models have been studied for a wide variety of natural Gibbs distributions and random graphical processes. Here, we establish feasiblity of random local access models for one of the most canonical such sample spaces, the set of satisfying assignments to a k-CNF formula.

Problem

• Random local access to k-SAT solutions
• Uniform distribution over satisfying assignments
• Sublinear time algorithm for k-CNF formulas

Innovation

• Sublinear time algorithm for k-SAT solutions
• Memory-less query access to assignments
• Emulates uniform distribution over satisfying assignments