A crucial task in the political redistricting problem is to sample redistricting plans i.e. a partitioning of the graph of census blocks into districts. We show that Recombination [DeFord-Duchin-Solomon'21]-a popular Markov chain to sample redistricting plans-is exponentially slow mixing on simple subgraph of $\mathbb{Z}_2.$ We show an alternative way to sample balance, compact and contiguous redistricting plans using a "relaxed" version of ReCom and rejection sampling.

翻译：政治选区重划问题中的一项关键任务是采样选区重划方案，即普查区块图划分为选区的方案。我们证明重组算法(Recombination [DeFord-Duchin-Solomon'21])——一种广泛用于采样选区重划方案的马尔可夫链——在 $\mathbb{Z}_2$ 的简单子图上具有指数级缓慢混合性。我们提出了一种替代方法，通过使用"松弛"版本的ReCom算法结合拒绝采样，来采样平衡、紧凑且连续的选区重划方案。