发表机构
Nanjing University; National Institute of Informatics（南京大学; 日本国立情报学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出计数型Lovász局部引理，针对一般约束满足问题，在特定局部引理条件下给出近似计数满足赋值的多项式时间算法，核心是新型2-树扩张，条件紧且匹配已知下界。
AI 中文摘要
我们建立了Lovász局部引理的计数类似物：针对一般约束满足问题（CSP），在局部引理条件$$4\textrm{e}\boldsymbol{\times}p\boldsymbol{\times}(D+1)^2 \boldsymbol{\times} 1$$下，给出了近似计数其满足赋值的多项式时间算法，其中$p$为最大约束违反概率，$D$为最大依赖度。该条件在常数因子范围内是紧的，与CSP自然子类中近似计数的已知下界$pD^2\boldsymbol{\times}1$匹配。我们方法的核心是针对约束边际概率的新型2-树扩张，其能捕捉局部引理 regime 下的相关性衰减。
英文摘要
We establish counting and sampling analogues of the Lovász Local Lemma: we give efficient algorithms for approximately counting and exactly sampling satisfying assignments of general constraint satisfaction problems (CSPs) in the local lemma regime $$4 \mathrm{e} p (D+1)^2\leq 1, $$ where $p$ is the maximum constraint violation probability and $D$ is the maximum dependency degree. This condition is tight up to constant factors under $\mathbf{NP}\neq\mathbf{RP}$, matching known hardness bounds for counting and sampling in natural subclasses of CSPs. Our key ingredient is a novel $2$-tree expansion for constraint marginal probabilities that exhibits exponential decay of correlations throughout this regime. This expansion yields deterministic polynomial-time approximate counting for fixed local parameters, randomized approximate counting with quadratic cost, and exact sampling in expected near-linear time when the local parameters are fixed.
CommentsV1 was titled "A Counting Lovász Local Lemma". V2 adds an exact sampler with expected near-linear running time under the same LLL condition. V3 substantially simplifies the construction of the near-linear time exact sampler
