The Lov\'asz Local Lemma is a versatile result in probability theory, characterizing circumstances in which a collection of $n$ `bad events', each occurring with probability at most $p$ and dependent on a set of underlying random variables, can be avoided. It is a central tool of the probabilistic method, since it can be used to show that combinatorial objects satisfying some desirable properties must exist. While the original proof was existential, subsequent work has shown algorithms for the Lov\'asz Local Lemma: that is, in circumstances in which the lemma proves the existence of some object, these algorithms can constructively find such an object. One main strand of these algorithms, which began with Moser and Tardos's well-known result (JACM 2010), involves iteratively resampling the dependent variables of satisfied bad events until none remain satisfied. In this paper, we present a novel analysis that can be applied to resampling-style Lov\'asz Local Lemma algorithms. This analysis shows that an output assignment for the dependent variables of most events can be determined only from $O(\log \log_{1/p} n)$-radius local neighborhoods, and that the events whose variables may still require resampling can be identified from these neighborhoods. This allows us to improve randomized complexities for the constructive Lov\'asz Local Lemma (with polynomial criterion) in several parallel and distributed models. In particular, we obtain: 1) A LOCAL algorithm with $O(\log\log_{1/p} n)$ node-averaged complexity (while matching the $O(\log_{1/p} n)$ worst-case complexity of Chung, Pettie, and Su). 2) An algorithm for the LCA and VOLUME models requiring $d^{O(\log\log_{1/p} n)}$ probes per query. 3) An $O(\log\log\log_{1/p} n)$-round algorithm for CONGESTED CLIQUE, linear space MPC, and Heterogenous MPC.

翻译：Lovász局部引理是概率论中的一个多用途结果，它刻画了当$n$个"坏事件"（每个事件发生的概率至多为$p$，且依赖于一组基础随机变量）可以被避免的情形。作为概率方法的核心工具，该引理可用于证明满足某些理想性质的组合对象必然存在。虽然原始证明是存在性的，后续研究已为Lovász局部引理提出了构造性算法：即在引理证明某对象存在的情形下，这些算法能实际构造出该对象。这类算法的主流方向始于Moser与Tardos的著名成果（JACM 2010），其核心思想是通过迭代重采样已满足坏事件的依赖变量，直至所有坏事件均不满足。本文提出一种适用于重采样式Lovász局部引理算法的新颖分析方法。该分析表明，对于大多数事件，其依赖变量的输出赋值仅需通过$O(\log \log_{1/p} n)$半径的局部邻域即可确定，且仍需重采样变量的事件亦可从这些邻域中识别。基于此，我们在多种并行与分布式模型中改进了构造性Lovász局部引理（满足多项式判据）的随机复杂度。具体而言，我们获得了：1）节点平均复杂度为$O(\log\log_{1/p} n)$的LOCAL算法（同时匹配Chung、Pettie与Su提出的$O(\log_{1/p} n)$最坏情况复杂度）；2）适用于LCA与VOLUME模型的算法，每次查询仅需$d^{O(\log\log_{1/p} n)}$次探测；3）在CONGESTED CLIQUE、线性空间MPC及异构MPC模型中实现$O(\log\log\log_{1/p} n)$轮复杂度的算法。