We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity , can be viewed as a strong generalization of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw-free graphs, which generalize line graphs. We consider two further generalizations of claw-free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially bounded vertex weights.

中文翻译：

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计算具有有界二分路径宽度的图中的独立集

我们展示了一个简单的马尔可夫链，即格劳伯动力学，可以在多项式时间内为某个类别的图有效地几乎均匀地随机采样独立集。该类由称为二分路径宽度的新图形参数的有界性确定。这个结果，我们证明了具有逸度的更一般的硬核分布 ，可以看作是 Jerrum 和 Sinclair 在近似计数匹配（即折线图中的独立集）方面的工作的强推广。具有有界二部路径宽度的图类包括无爪图，它概括了线图。我们考虑了无爪图的两个进一步概括，并证明这些类具有有界的二部路径宽度。我们还展示了如何将所有结果扩展到多项式有界顶点权重。