Fast distributed algorithms that output a feasible solution for constraint satisfaction problems, such as maximal independent sets, have been heavily studied. There has been much less research on distributed sampling problems, where one wants to sample from a distribution over all feasible solutions (e.g., uniformly sampling a feasible solution). Recent work (Feng, Sun, Yin PODC 2017; Fischer and Ghaffari DISC 2018; Feng, Hayes, and Yin arXiv 2018) has shown that for some constraint satisfaction problems there are distributed Markov chains that mix in $O(\log n)$ rounds in the classical LOCAL model of distributed computation. However, these methods return samples from a distribution close to the desired distribution, but with some small amount of error. In this paper, we focus on the problem of exact distributed sampling. Our main contribution is to show that these distributed Markov chains in tandem with techniques from the sequential setting, namely coupling from the past and bounding chains, can be used to design $O(\log n)$-round LOCAL model exact sampling algorithms for a class of weighted local constraint satisfaction problems. This general result leads to $O(\log n)$-round exact sampling algorithms that use small messages (i.e., run in the CONGEST model) and polynomial-time local computation for some important special cases, such as sampling weighted independent sets (aka the hardcore model) and weighted dominating sets.

翻译：摘要：针对约束满足问题（如极大独立集）输出可行解的快速分布式算法已被广泛研究。然而，关于分布式采样问题的研究相对较少——这类问题需要从所有可行解的分布中采样（例如，均匀采样可行解）。近期工作（Feng, Sun, Yin PODC 2017; Fischer and Ghaffari DISC 2018; Feng, Hayes, and Yin arXiv 2018）表明，对于某些约束满足问题，存在分布式马尔可夫链能够在经典LOCAL分布式计算模型中以$O(\log n)$轮混合。然而，这些方法返回的样本来自接近目标分布的分布，但存在少量误差。本文聚焦于精确分布式采样问题。我们的主要贡献在于证明：将此类分布式马尔可夫链与序贯设置中的技术（即"来自过去的耦合"与"边界链"）相结合，可为加权局部约束满足问题设计出$O(\log n)$轮LOCAL模型精确采样算法。这一通用结果催生出若干重要特例的$O(\log n)$轮精确采样算法，这些算法使用小消息（即在CONGEST模型中运行），并具有多项式时间的局部计算能力，例如加权独立集采样（即硬核模型）与加权支配集采样。