We give a new rapid mixing result for a natural random walk on the independent sets of a graph $G$. We show that when $G$ has bounded treewidth, this random walk -- known as the Glauber dynamics for the hardcore model -- mixes rapidly for all fixed values of the standard parameter $\lambda > 0$, giving a simple alternative to existing sampling algorithms for these structures. We also show rapid mixing for analogous Markov chains on dominating sets, $b$-edge covers, $b$-matchings, maximal independent sets, and maximal $b$-matchings. (For $b$-matchings, maximal independent sets, and maximal $b$-matchings we also require bounded degree.) Our results imply simpler alternatives to known algorithms for the sampling and approximate counting problems in these graphs. We prove our results by applying a divide-and-conquer framework we developed in a previous paper, as an alternative to the projection-restriction technique introduced by Jerrum, Son, Tetali, and Vigoda. We extend this prior framework to handle chains for which the application of that framework is not straightforward, strengthening existing results by Dyer, Goldberg, and Jerrum and by Heinrich for the Glauber dynamics on $q$-colorings of graphs of bounded treewidth and bounded degree.

翻译：本文针对图$G$的独立集上的自然随机游走给出了一个新的快速混合结果。我们证明，当$G$具有有界树宽时，这种随机游走（即硬核模型的Glauber动力学）对于标准参数$\lambda > 0$的所有固定值均能快速混合，为这些结构上的现有采样算法提供了一种简单的替代方案。我们还证明了支配集、$b$-边覆盖、$b$-匹配、极大独立集和极大$b$-匹配上的类似马尔可夫链也具有快速混合性（对于$b$-匹配、极大独立集和极大$b$-匹配，还需有界度条件）。我们的结果为这些图中的采样和近似计数问题提供了比已知算法更简单的替代方案。我们通过应用先前论文中开发的"分而治之"框架来证明结果，这替代了Jerrum、Son、Tetali和Vigoda引入的投影-限制技术。我们扩展了这一先前框架，以处理该框架应用不直接的链，从而强化了Dyer、Goldberg和Jerrum以及Heinrich关于有界树宽和有界度图上$q$-着色Glauber动力学的现有结果。