采样连通图划分的复杂性和几何性
Complexity and Geometry of Sampling Connected Graph Partitions
摘要：
在本文中，我们证明了棘手的结果采样从一个平面图的分区到连通组件的集合。我们的证明是由Jerrum，Valiant和Vazirani介绍的技术的动机。此外，我们使用的小工具的启发，他们的技术，以提供家庭的图形，在实践中使用的“翻转行走”马尔可夫链的采样任务表现出指数缓慢的混合。支持我们的理论结果，我们提出了一些实证证据，证明了缓慢混合的翻转行走网格图和真实数据。受自避免行走的统计物理学的启发，我们研究了某些流行的采样算法对图拓扑结构的敏感性。最后，我们讨论了几种情况下，抽样问题是听话的。政治选区重划的应用程序最近带来了越来越多的关注这个问题，我们阐明了开放的问题，我们的结果突出显示了这个应用程序。
英文摘要：
In this paper, we prove intractability results about sampling from the set of partitions of a planar graph into connected components. Our proofs are motivated by a technique introduced by Jerrum, Valiant, and Vazirani. Moreover, we use gadgets inspired by their technique to provide families of graphs where the "flip walk" Markov chain used in practice for this sampling task exhibits exponentially slow mixing. Supporting our theoretical results we present some empirical evidence demonstrating the slow mixing of the flip walk on grid graphs and on real data. Inspired by connections to the statistical physics of self-avoiding walks, we investigate the sensitivity of certain popular sampling algorithms to the graph topology. Finally, we discuss a few cases where the sampling problem is tractable. Applications to political redistricting have recently brought increased attention to this problem, and we articulate open questions about this application that are highlighted by our results.
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