The Lovász Local Lemma (LLL) shows that, for a collection of “bad” events B in a probability space that are not too likely and not too interdependent, there is a positive probability that no events in B occur. Moser and Tardos (2010) gave sequential and parallel algorithms that transformed most applications of the variable-assignment LLL into efficient algorithms. A framework of Harvey and Vondrák (2015) based on “resampling oracles” extended this to sequential algorithms for other probability spaces satisfying a generalization of the LLL known as the Lopsided Lovász Local Lemma (LLLL). We describe a new structural property that holds for most known resampling oracles, which we call “obliviousness.” Essentially, it means that the interaction between two bad-events B , B ′ depends only on the randomness used to resample B and not the precise state within B itself. This property has two major consequences. First, combined with a framework of Kolmogorov (2016), it leads to a unified parallel LLLL algorithm, which is faster than previous, problem-specific algorithms of Harris (2016) for the variable-assignment LLLL and of Harris and Srinivasan (2014) for permutations. This gives the first RNC algorithms for rainbow perfect matchings and rainbow Hamiltonian cycles of K n . Second, this property allows us to build LLLL probability spaces from simpler “atomic” events. This gives the first resampling oracle for rainbow perfect matchings on the complete s -uniform hypergraph K n ( s ) and the first commutative resampling oracle for Hamiltonian cycles of K n .

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不经意的重采样预言机和不平衡 Lovász 局部引理的并行算法

Lovász 局部引理 (LLL) 表明，对于概率空间中不太可能且不太相互依赖的“坏”事件 B 的集合，B 中没有事件发生的概率为正。Moser 和 Tardos (2010) 给出了顺序和并行算法，这些算法将变量赋值 LLL 的大多数应用转化为高效算法。Harvey 和 Vondrák (2015) 的一个基于“重采样预言”的框架将其扩展到其他概率空间的顺序算法，满足 LLL 的泛化，称为 Lopsid Lovász 局部引理 (LLLL)。我们描述了一种新的结构特性，它适用于大多数已知的重采样预言机，我们称之为“遗忘”。本质上，它意味着两个坏事件之间的交互乙,乙 '仅取决于用于重采样的随机性乙而不是内部的精确状态乙本身。该属性有两个主要后果。首先，结合 Kolmogorov (2016) 的框架，它产生了一个统一的并行 LLLL 算法，该算法比 Harris (2016) 的变量分配 LLLL 以及 Harris 和 Srinivasan (2014) 的针对特定问题的算法更快。为排列。这给出了第一个用于彩虹完美匹配和彩虹哈密顿循环的 RNC 算法ķ n . 其次，这个属性允许我们从更简单的“原子”事件构建 LLLL 概率空间。这为完整的彩虹完美匹配提供了第一个重采样预言机s- 均匀超图ķ n (s)和哈密顿循环的第一个交换重采样预言ķ n .