Fast Almost-Uniform Sampling of Random $k$-SAT Solutions

2026-10-07

AI Summary

This study addresses the problem of approximately uniform sampling of satisfying assignments for random $k$-SAT formulas. To this end, it proposes an efficient polynomial-time sampling algorithm that decouples the high-degree core from residual variables, which are then sampled recursively. The approach integrates a recursive insertion chain framework with approximate block-Glauber dynamics updates and polymer expansion techniques. This work surpasses traditional density limitations by achieving a universal polynomial runtime independent of both clause width and formula density. At specific densities, the proposed algorithm significantly outperforms existing methods while providing rigorous guarantees that the output distribution closely approximates the uniform distribution with controllable computational overhead.

Abstract

We study approximately uniform sampling of satisfying assignments from random $k$-SAT formulas. For every sufficiently large $k$ and density $0 < α\le 2^k/k^{16}$, we prove that, with high probability over the formula, there is a sampler whose output distribution is within total variation distance $\varepsilon$ of the uniform distribution on satisfying assignments and whose expected running time is at most $(nk(α+1)/\varepsilon)^C$, for a universal constant $C$. Our algorithm improves the counting and sampling algorithms obtained by Chen, Lonkar, Wang, Yang, and Yin (STOC 2025) at the density $2^k/\operatorname{poly}(k)$ with running time $(n/\varepsilon)^{\operatorname{poly}(k,α)}$. Our result achieves this density region for sampling with a polynomial degree independent of both the width and the density. Our algorithm separates a high-degree core from the remaining variables, and combines a recursive sampler for the residual formulas with approximate block heat-bath updates on the core. We adapt the recursive insertion-chain framework of Jain, Mizgerd, and Pham (2026) from $2$-trees to ordinary connected violation sets. Expansion and random literal signs yield uniform moment bounds for the resulting correlated lists across all residual formulas, allowing the signed-flow analysis to give a universal polynomial running-time degree. A polymer expansion and an exploration bound establish a polynomial spectral gap for the core dynamics.

Problem

• random k-SAT
• almost-uniform sampling
• satisfying assignments
• approximate counting

Innovation

• random k-SAT sampling
• recursive insertion-chain framework
• block heat-bath updates
• spectral gap
• polymer expansion