Let $\Phi = (V, \mathcal{C})$ be a constraint satisfaction problem on variables $v_1,\dots, v_n$ such that each constraint depends on at most $k$ variables and such that each variable assumes values in an alphabet of size at most $[q]$. Suppose that each constraint shares variables with at most $\Delta$ constraints and that each constraint is violated with probability at most $p$ (under the product measure on its variables). We show that for $k, q = O(1)$, there is a deterministic, polynomial time algorithm to approximately count the number of satisfying assignments and a randomized, polynomial time algorithm to sample from approximately the uniform distribution on satisfying assignments, provided that \[C\cdot q^{3}\cdot k \cdot p \cdot \Delta^{7} < 1, \quad \text{where }C \text{ is an absolute constant.}\] Previously, a result of this form was known essentially only in the special case when each constraint is violated by exactly one assignment to its variables. For the special case of $k$-CNF formulas, the term $\Delta^{7}$ improves the previously best known $\Delta^{60}$ for deterministic algorithms [Moitra, J.ACM, 2019] and $\Delta^{13}$ for randomized algorithms [Feng et al., arXiv, 2020]. For the special case of properly $q$-coloring $k$-uniform hypergraphs, the term $\Delta^{7}$ improves the previously best known $\Delta^{14}$ for deterministic algorithms [Guo et al., SICOMP, 2019] and $\Delta^{9}$ for randomized algorithms [Feng et al., arXiv, 2020].

中文翻译：

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迈向抽样的Lovász地方引理

令$ \ Phi =（V，\ mathcal {C}）$是变量$ v_1，\ dots，v_n $的约束满足问题，这样每个约束最多取决于$ k $个变量，并且每个变量都采用大小不超过$ [q] $的字母。假设每个约束共享最多具有$ \ Delta $约束的变量，并且每个约束具有至多$ p $的概率被违反（在其变量的乘积度量下）。我们显示出对于$ k，q = O（1）$，存在一种确定性的多项式时间算法来近似计算满足分配的数量，以及一种随机的多项式时间算法来从满足分配的近似均匀分布中进行采样\\ [C \ cdot q ^ {3} \ cdot k \ cdot p \ cdot \ Delta ^ {7} <1，\ quad \ text {其中} C \ text {是绝对常数。} \]以前，这种形式的结果本质上仅在特殊情况下才知道，即在每种情况下，只有对其变量的一个分配就违反了每个约束。对于$ k $ -CNF公式的特殊情况，术语$ \ Delta ^ {7} $对确定性算法[Moitra，J.ACM，2019]和$ \改进了以前最著名的$ \ Delta ^ {60} $。 Delta ^ {13} $用于随机算法[Feng等人，arXiv，2020年]。对于适当地为$ q $着色$ k $均匀超图的特殊情况，术语$ \ Delta ^ {7} $改进了确定性算法中先前最知名的$ \ Delta ^ {14} $ [Guo等， SICOMP，2019年]和$ \ Delta ^ {9} $用于随机算法[Feng等人，arXiv，2020年]。对于确定性算法[Moitra，J.ACM，2019]，术语$ \ Delta ^ {7} $改进了先前最著名的$ \ Delta ^ {60} $，对随机算法则改进了$ \ Delta ^ {13} $ [Feng等等人，arXiv，2020年]。对于适当地为$ q $着色$ k $均匀超图的特殊情况，术语$ \ Delta ^ {7} $改进了确定性算法中先前最知名的$ \ Delta ^ {14} $ [Guo等， SICOMP，2019年]和$ \ Delta ^ {9} $用于随机算法[Feng等人，arXiv，2020年]。对于确定性算法[Moitra，J.ACM，2019]，术语$ \ Delta ^ {7} $改进了先前最著名的$ \ Delta ^ {60} $，对随机算法则改进了$ \ Delta ^ {13} $ [Feng等等人，arXiv，2020年]。对于适当地为$ q $着色$ k $均匀超图的特殊情况，术语$ \ Delta ^ {7} $改进了确定性算法中先前最知名的$ \ Delta ^ {14} $ [Guo等， SICOMP，2019年]和$ \ Delta ^ {9} $用于随机算法[Feng等人，arXiv，2020年]。