Random sampling of graph partitions under constraints has become a popular tool for evaluating legislative redistricting plans. Analysts detect partisan gerrymandering by comparing a proposed redistricting plan with an ensemble of sampled alternative plans. For successful application, sampling methods must scale to maps with a moderate or large number of districts, incorporate realistic legal constraints, and accurately and efficiently sample from a selected target distribution. Unfortunately, most existing methods struggle in at least one of these areas. We present a new Sequential Monte Carlo (SMC) algorithm that generates a sample of redistricting plans converging to a realistic target distribution. Because it draws many plans in parallel, the SMC algorithm can efficiently explore the relevant space of redistricting plans better than the existing Markov chain Monte Carlo (MCMC) algorithms that generate plans sequentially. Our algorithm can simultaneously incorporate several constraints commonly imposed in real-world redistricting problems, including equal population, compactness, and preservation of administrative boundaries. We validate the accuracy of the proposed algorithm by using a small map where all redistricting plans can be enumerated. We then apply the SMC algorithm to evaluate the partisan implications of several maps submitted by relevant parties in a recent high-profile redistricting case in the state of Pennsylvania. We find that the proposed algorithm converges faster and with fewer samples than a comparable MCMC algorithm. Open-source software is available for implementing the proposed methodology.

翻译：在约束条件下对图划分进行随机采样已成为评估立法选区划分方案的热门工具。分析人员通过将拟议的选区划分方案与采样得到的备选方案集合进行比较，来检测党派不公正划分选区行为。为成功应用，采样方法必须能扩展至包含中等或大量选区的地图，纳入现实法律约束，并准确高效地从选定的目标分布中进行采样。遗憾的是，现有方法大多至少在一个方面存在不足。我们提出了一种新的序贯蒙特卡洛（SMC）算法，该算法能生成收敛至真实目标分布的选区划分方案样本。由于可并行生成多个方案，SMC算法能比现有顺序生成方案的马尔可夫链蒙特卡洛（MCMC）算法更高效地探索选区划分方案的相关空间。我们的算法可同时纳入现实选区划分问题中常见的多种约束条件，包括人口均衡、紧凑性以及行政边界保留。通过使用可枚举所有选区划分方案的小型地图，我们验证了所提出算法的准确性。随后，我们应用SMC算法评估了宾夕法尼亚州近期备受关注的选区划分案件中相关方提交的若干地图的党派影响。研究发现，与同类MCMC算法相比，所提出算法收敛更快且所需样本更少。实现该方法的开源软件现已可获取。