When auditing a redistricting plan, a persuasive method is to compare the plan with an ensemble of neutrally drawn redistricting plans. Ensembles are generated via algorithms that sample distributions on balanced graph partitions. To audit the partisan difference between the ensemble and a given plan, one must ensure that the non-partisan criteria are matched so that we may conclude that partisan differences come from bias rather than, for example, levels of compactness or differences in community preservation. Certain sampling algorithms allow one to explicitly state the policy-based probability distribution on plans, however, these algorithms have shown poor mixing times for large graphs (i.e. redistricting spaces) for all but a few specialized measures. In this work, we generate a multiscale parallel tempering approach that makes local moves at each scale. The local moves allow us to adopt a wide variety of policy-based measures. We examine our method in the state of Connecticut and succeed at achieving fast mixing on a policy-based distribution that has never before been sampled at this scale. Our algorithm shows promise to expand to a significantly wider class of measures that will (i) allow for more principled and situation-based comparisons and (ii) probe for the typical partisan impact that policy can have on redistricting.

翻译：在审计选区重划计划时，一种具有说服力的方法是将该计划与一组中立绘制的选区重划计划集合进行比较。该集合通过采样平衡图划分分布的算法生成。为审计该集合与给定计划之间的党派差异，必须确保非党派标准保持一致，从而能够得出党派差异源于偏见而非紧凑性程度或社区保留差异等要素的结论。部分采样算法允许明确设定基于政策的计划概率分布，但此类算法在处理大规模图（即选区重划空间）时，除少数特定度量外，均表现出较差的混合时间。本研究提出一种多尺度并行回火方法，在每个尺度上执行局部移动。局部移动使我们能够采用多样化的基于政策的度量。我们在康涅狄格州验证该方法，成功实现了此前从未在该尺度上采样的基于政策分布的快速混合。该算法有望扩展至更广泛的度量类别，从而（i）支持更符合原则且基于情境的比较，（ii）探究政策对选区重划可能产生的典型党派影响。