2024/11/05 by Zongchen Chen, Aditya Lonkar, Chen, Zongchen +7
Computer Science · #Bayesian Modeling and Causal Inference #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Logic, Reasoning, and Knowledge #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2411.02980
openalex publication_date 2024/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present efficient counting and sampling algorithms for random k-SAT when the clause density satisfies α≤ (2k)/(poly(k)). In particular, the exponential term 2k matches the satisfiability threshold Θ(2k) for the existence of a solution and the (conjectured) algorithmic threshold 2k (ln k) / k for efficiently finding a solution. Previously, the best-known counting and sampling algorithms required far more restricted densities α\lesssim 2k/3 [He, Wu, Yang, SODA '23]. Notably, our result goes beyond the lower bound d\gtrsim 2k/2 for worst-case k-SAT with bounded-degree d [Bezáková et al, SICOMP '19], showing that for counting and sampling, the average-case random k-SAT model is computationally much easier than the worst-case model. At the heart of our approach is a new refined analysis of the recent novel coupling procedure by [Wang, Yin, FOCS '24], utilizing the structural properties of random constraint satisfaction problems (CSPs). Crucially, our analysis avoids reliance on the 2-tree structure used in prior works, which cannot extend beyond the worst-case threshold 2k/2. Instead, we employ a witness tree similar to that used in the analysis of the Moser-Tardos algorithm [Moser, Tardos, JACM '10] for the Lovász Local lemma, which may be of independent interest. Our new analysis provides a universal framework for efficient counting and sampling for random atomic CSPs, including, for example, random hypergraph colorings. At the same time, it immediately implies as corollaries several structural and probabilistic properties of random CSPs that have been widely studied but rarely justified, including replica symmetry and non-reconstruction.