发表机构
Renmin University of China; Shanghai Jiao Tong University（中国人民大学; 上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对随机 $k$-SAT 公式，提出一种近似均匀采样满足赋值的算法，在密度 $2^k/k^{16}$ 以下实现与宽度和密度无关的多项式运行时间，改进了现有方法。
AI 中文摘要
我们研究从随机 $k$-SAT 公式中对满足赋值进行近似均匀采样的问题。对于每个足够大的 $k$ 和密度 $0 < \alpha \le 2^k/k^{16}$，我们证明，以高概率（关于公式）存在一个采样器，其输出分布与满足赋值上的均匀分布的总变差距离在 $\varepsilon$ 以内，并且其期望运行时间至多为 $(nk(\alpha+1)/\varepsilon)^C$，其中 $C$ 是一个通用常数。我们的算法改进了 Chen、Lonkar、Wang、Yang 和 Yin（STOC 2025）在密度 $2^k/\operatorname{poly}(k)$ 下获得的计数和采样算法，其运行时间为 $(n/\varepsilon)^{\operatorname{poly}(k,\alpha)}$。我们的结果在采样方面达到了该密度区域，且多项式次数与宽度和密度均无关。我们的算法将高度数核心与其余变量分离，并将剩余公式的递归采样器与核心上的近似块热浴更新相结合。我们将 Jain、Mizgerd 和 Pham（2026）的递归插入链框架从 $2$-树扩展到普通连通违规集。扩展性和随机文字符号为所有剩余公式中产生的相关列表提供了均匀矩界，使得符号流分析能够给出通用的多项式运行时间次数。聚合物展开和探索界为核心动力学建立了多项式谱间隙。
英文摘要
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.
