oblivious 重采样预言机与 Lopsided Lovász Local Lemma 的并行算法
Oblivious resampling oracles and parallel algorithms for the Lopsided Lovasz Local Lemma
AI总结：
本文提出重采样预言机的 oblivious 性质，据此实现统一的并行 LLLL 算法，首次给出彩虹完美匹配和彩虹哈密顿圈的 RNC 算法，并构建了相应的顺序与交换重采样预言机。
摘要：
Lovász Local Lemma (LLL) 是一个概率工具，它表明：如果概率空间中的一组“坏”事件 $\mathcal B$ 不太可能发生且相互依赖程度不高，那么存在正概率使得 $\mathcal B$ 中没有坏事件发生。Moser & Tardos (2010) 给出了顺序和并行算法，将变量赋值 LLL 的大多数应用转化为高效算法。Harvey & Vondrák (2015) 基于“重采样预言机”的框架将其推广到满足 Lopsided Lovász Local Lemma (LLLL) 的其他概率空间的一般顺序算法。\n我们描述了一个新的结构性质，该性质对所有已知的重采样预言机都成立，我们称之为“obliviousness”（无意识性）。本质上，它意味着两个坏事件 $B, B'$ 之间的相互作用仅取决于用于重采样 $B$ 的随机性，而不取决于 $B$ 内部的具体状态。\n这一性质有两个主要后果。首先，结合 Kolmogorov (2016) 的框架，它是实现统一的并行 LLLL 算法的关键，该算法比之前针对变量赋值 LLLL 算法的 Harris (2016) 和针对排列的 Harris & Srinivasan (2014) 的问题特定算法更快。这给出了 $K_n$ 的彩虹完美匹配和彩虹哈密顿圈的首次 RNC 算法。\n其次，这一性质允许我们从相对简单的“原子”事件构建 LLLL 概率空间。这提供了完整 $s$-均匀超图 $K_n^{(s)}$ 上彩虹完美匹配的第一个顺序重采样预言机，以及 $K_n$ 的哈密顿圈的第一个交换重采样预言机。
英文摘要：
The Lovász Local Lemma (LLL) is a probabilistic tool which shows that, if a collection of "bad" events $\mathcal B$ in a probability space are not too likely and not too interdependent, then there is a positive probability that no bad-events in $\mathcal B$ occur. Moser & Tardos (2010) gave sequential and parallel algorithms which transformed most applications of the variable-assignment LLL into efficient algorithms. A framework of Harvey & Vondrák (2015) based on "resampling oracles" extended this to general sequential algorithms for other probability spaces satisfying the Lopsided Lovász Local Lemma (LLLL). We describe a new structural property which holds for all 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 is the key to achieving a unified parallel LLLL algorithm, which is faster than previous, problem-specific algorithms of Harris (2016) for the variable-assignment LLLL algorithm and of Harris \& 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 out of relatively simple "atomic" events. This provides the first sequential 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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