Markov Chain Monte Carlo (MCMC) algorithms are a widely-used algorithmic tool for sampling from high-dimensional distributions, a notable example is the equilibirum distribution of graphical models. The Glauber dynamics, also known as the Gibbs sampler, is the simplest example of an MCMC algorithm; the transitions of the chain update the configuration at a randomly chosen coordinate at each step. Several works have studied distributed versions of the Glauber dynamics and we extend these efforts to a more general family of Markov chains. An important combinatorial problem in the study of MCMC algorithms is random colorings. Given a graph $G$ of maximum degree $\Delta$ and an integer $k\geq\Delta+1$, the goal is to generate a random proper vertex $k$-coloring of $G$. Jerrum (1995) proved that the Glauber dynamics has $O(n\log{n})$ mixing time when $k>2\Delta$. Fischer and Ghaffari (2018), and independently Feng, Hayes, and Yin (2018), presented a parallel and distributed version of the Glauber dynamics which converges in $O(\log{n})$ rounds for $k>(2+\varepsilon)\Delta$ for any $\varepsilon>0$. We improve this result to $k>(11/6-\delta)\Delta$ for a fixed $\delta>0$. This matches the state of the art for randomly sampling colorings of general graphs in the sequential setting. Whereas previous works focused on distributed variants of the Glauber dynamics, our work presents a parallel and distributed version of the more general flip dynamics presented by Vigoda (2000) (and refined by Chen, Delcourt, Moitra, Perarnau, and Postle (2019)), which recolors local maximal two-colored components in each step.

翻译：马尔可夫链蒙特卡洛（MCMC）算法是从高维分布（典型例子是图模型的均衡分布）中进行采样的广泛使用的算法工具。Glauber动力学（也称为吉布斯采样器）是MCMC算法中最简单的例子：该链的转移步骤在每个时刻随机选取一个坐标并更新其配置。已有若干研究探讨了Glauber动力学的分布式版本，而我们将这些工作扩展到更一般的马尔可夫链族。MCMC算法研究中的一个重要组合问题是随机着色。给定一个最大度为$\Delta$的图$G$和一个整数$k\geq\Delta+1$，目标是生成$G$的一个随机正常顶点$k$着色。Jerrum（1995）证明，当$k>2\Delta$时，Glauber动力学的混合时间为$O(n\log{n})$。Fischer和Ghaffari（2018），以及Feng、Hayes和Yin（2018）独立提出了Glauber动力学的并行与分布式版本，该版本在$k>(2+\varepsilon)\Delta$（对任意$\varepsilon>0$）时可在$O(\log{n})$轮内收敛。我们将此结果改进为$k>(11/6-\delta)\Delta$，其中$\delta>0$为固定常数。这一结果与序列设置下一般图随机着色的当前最优水平持平。与先前工作聚焦于Glauber动力学的分布式变体不同，我们的工作提出了由Vigoda（2000）提出（并由Chen、Delcourt、Moitra、Perarnau和Postle（2019）改进）的更一般的翻转动力学（flip dynamics）的并行与分布式版本，该动力学在每一步中对局部最大二色分量进行重新着色。