We consider the irreducibility of switch-based Markov chains for the approximate uniform sampling of Hamiltonian cycles in a given undirected dense graph on $n$ vertices. As our main result, we show that every pair of Hamiltonian cycles in a graph with minimum degree at least $n/2+7$ can be transformed into each other by switch operations of size at most 10, implying that the switch Markov chain using switches of size at most 10 is irreducible. As a proof of concept, we also show that this Markov chain is rapidly mixing on dense monotone graphs.
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机译:我们考虑基于交换机的马尔可夫链条的不可缩写,以便在$ N $顶点上的给定的无向密集图中的哈密顿周期均匀采样。作为我们的主要结果,我们表明,最低限度至少$ N / 2 + 7 $的图表中的每对哈密顿周期可以通过最多10个尺寸的切换操作来互相转换,这意味着交换机马尔可夫链使用大多数10的尺寸交换机是不可缩短的。作为概念证明,我们还表明,这个马尔可夫链在致密的单调图上迅速混合。
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