ReCom is a leading Markov Chain Monte Carlo algorithm for sampling balanced graph partitions in computational redistricting. At each step, its transition function proposes a new partition by merging two adjacent districts and if possible re-splitting the conjoined region. The transition function is efficient in practice, however, it is unknown whether it is guaranteed to run in polynomial time. In this report we exhibit an explicit family of 3-partitions on planar square grid graphs from which ReCom requires an exponentially large expected number of steps to re-split the graph (even if we admit approximately balanced splits), showing that in the worst case ReCom does not run in polynomial time. Notably, this result implies that ReCom is not technically rapidly mixing (if started from an adversarial configuration, ReCom requires exponential many steps to reach the stationary distribution).

翻译：ReCom是在计算选区重划中用于采样平衡图划分的领先马尔可夫链蒙特卡罗算法。在每一步中，其转移函数通过合并两个相邻区域，并在可能的情况下重新分割合并区域，来提出新的划分。该转移函数在实践中高效运作，然而，尚未证明其是否保证在多项式时间内运行。本报告中，我们在平面方形网格图上展示了一个显式的三划分族，从该族出发，ReCom需要指数级期望步数才能重新分割图（即使我们允许近似平衡的划分），这表明在最坏情况下ReCom并非在多项式时间内运行。值得注意的是，该结果意味着ReCom在技术上不具备快速混合性（若从对抗性配置出发，ReCom需要指数级步数才能达到平稳分布）。