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：马尔可夫链蒙特卡罗（MCMC）算法是一种广泛使用的从高维分布中采样的算法工具，一个显著的例子是图模型的平衡分布。Glauber动态（也称为Gibbs采样器）是MCMC算法中最简单的实例；该链的转移过程在每一步随机选择一个坐标更新配置。已有若干研究工作探讨了Glauber动态的分布式版本，我们将这些努力拓展至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。给定最大度为$\Delta$的图$G$与整数$k\geq\Delta+1$，目标是生成$G$的一个随机真顶点$k$-着色。Jerrum（1995）证明了当$k>2\Delta$时Glauber动态具有$O(n\log{n})$混合时间。Fischer与Ghaffari（2018），以及Feng、Hayes和Yin（2018）分别独立提出了Glauber动态的并行分布式版本，该版本在$k>(2+\varepsilon)\Delta$（对于任意$\varepsilon>0$）条件下可在$O(\log{n})$轮内收敛。我们将此结果改进至$k>(11/6-\delta)\Delta$（其中$\delta>0$为固定值）。这匹配了在顺序设置下对一般图随机采样着色的最新技术水平。先前研究主要关注Glauber动态的分布式变体，而我们的工作提出了由Vigoda（2000）提出（并经Chen、Delcourt、Moitra、Perarnau和Postle（2019）完善）的更一般的翻转动态的并行分布式版本，该动态在每一步对局部最大的双色连通分量进行重新着色。