The Lov\'{a}sz Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection $\mathcal B$ of "bad" events which are mostly independent and have low probability. In its simplest "symmetric" form, it asserts that whenever a bad-event has probability $p$ and affects at most $d$ bad-events, and $e p d < 1$, then a configuration avoiding all $\mathcal B$ exists. A seminal algorithm of Moser & Tardos (2010) gives nearly-automatic randomized algorithms for most constructions based on the LLL. However, deterministic algorithms have lagged behind. We address three specific shortcomings of the prior deterministic algorithms. First, our algorithm applies to the LLL criterion of Shearer (1985); this is more powerful than alternate LLL criteria and also removes a number of nuisance parameters and leads to cleaner and more legible bounds. Second, we provide parallel algorithms with much greater flexibility in the functional form of of the bad-events. Third, we provide a derandomized version of the MT-distribution, that is, the distribution of the variables at the termination of the MT algorithm. We show applications to non-repetitive vertex coloring, independent transversals, strong coloring, and other problems. These give deterministic algorithms which essentially match the best previous randomized sequential and parallel algorithms.

翻译：Lovász局部引理(LLL)是概率论中的基石性原理，它保证存在一种配置能够避开一组“坏”事件$\mathcal B$，这些事件基本相互独立且概率较低。在其最简单的“对称”形式中，该引理断言：当坏事件的概率为$p$且最多影响$d$个坏事件，且$epd<1$时，则存在一种能避开所有$\mathcal B$的配置。Moser与Tardos(2010)的开创性算法为基于LLL的大多数构造提供了近乎自动的随机算法。然而，确定性算法的发展相对滞后。本文旨在解决先前确定性算法的三个具体缺陷。首先，我们的算法适用于Shearer(1985)提出的LLL准则；该准则比其他LLL准则更具普适性，同时消除了多个冗余参数，使界限更清晰、更易解读。其次，我们提供了并行算法，在坏事件函数形式方面具有更大的灵活性。第三，我们给出了MT分布（即MT算法终止时变量的分布）的去随机化版本。我们展示了该方法在非重复顶点着色、独立横贯、强着色以及其他问题中的应用。这些确定性算法在性能上基本与先前最优的随机序贯算法和并行算法相当。