The process of drawing electoral district boundaries is known as political redistricting. Within this context, gerrymandering is the practice of drawing these boundaries such that they unfairly favor a particular political party, often leading to unequal representation and skewed electoral outcomes. One of the few ways to detect gerrymandering is by algorithmically sampling redistricting plans. Previous methods mainly focus on sampling from some neighborhood of ``realistic' districting plans, rather than a uniform sample of the entire space. We present a deterministic subexponential time algorithm to uniformly sample from the space of all possible $ k $-partitions of a bounded degree planar graph, and with this construct a sample of the entire space of redistricting plans. We also give a way to restrict this sample space to plans that match certain compactness and population constraints at the cost of added complexity. The algorithm runs in $ 2^{O(\sqrt{n}\log n)} $ time, although we only give a heuristic implementation. Our method generalizes an algorithm to count self-avoiding walks on a square to count paths that split general planar graphs into $ k $ regions, and uses this to sample from the space of all $ k $-partitions of a planar graph.

翻译：选举区域边界的绘制过程被称为政治选区重划。在此背景下，杰利蝾螈（gerrymandering）指的是通过绘制边界不公正地偏袒某一特定政党的做法，这常导致代表性不平等和选举结果扭曲。检测杰利蝾螈的少数方法之一是通过算法对选区规划方案进行采样。以往的方法主要聚焦于从“现实”选区规划方案的某个邻域中采样，而非对整个空间进行均匀采样。我们提出一种确定性的亚指数时间算法，用于对有界度平面图的所有可能$k$-分割空间进行均匀采样，并据此构建整个选区规划方案空间的样本。同时，我们提供了一种以增加复杂度为代价，将该样本空间限制为符合特定紧凑性和人口约束的方案的方法。该算法的运行时间为$2^{O(\sqrt{n}\log n)}$，尽管我们仅给出启发式实现。我们的方法将用于计数正方形上自避行走的算法推广为计数将一般平面图分割为$k$个区域的路径，并利用此方法对平面图的所有$k$-分割空间进行采样。