Partisan gerrymandering is a major cause for voter disenfranchisement in United States. However, convincing US courts to adopt specific measures to quantify gerrymandering has been of limited success to date. Recently, Stephanopoulos and McGhee in several papers introduced a new measure of partisan gerrymandering via the so-called “efficiency gap” that computes the absolute difference of wasted votes between two political parties in a two-party system; from a legal point of view the measure was found legally convincing in a US appeals court in a case that claims that the legislative map of the state of Wisconsin was gerrymandered. The goal of this article is to formalize and provide theoretical and empirical algorithmic analyses of the computational problem of minimizing this measure. To this effect, we show the following: (a) On the theoretical side, we formalize the corresponding minimization problem and provide non-trivial mathematical and computational complexity properties of the problem of minimizing the efficiency gap measure. Specifically, we prove the following results for the formalized minimization problem: (i) We show that the efficiency gap measure attains only a finite discrete set of rational values. (observations of similar nature but using different arguments were also made independently by Cho and Wendy (Univ Pa Law Rev 166(1), Article 2, 2017). (ii) We show that, assuming P \(\ne \textsf {NP}\), for general maps and arbitrary numeric electoral data the minimization problem does not admit any polynomial time algorithm with finite approximation ratio. Moreover, we show that the problem still remains \(\textsf {NP}\)-complete even if the numeric electoral data is linear in the number of districts, provided the map is provided in the form of a planar graph (or, equivalently, a polygonal subdivision of the two-dimensional Euclidean plane). (iii) Notwithstanding the previous hardness results, we show that efficient exact or efficient approximation algorithms can be designed if one assumes some reasonable restrictions on the map and electoral data. Items (ii) and (iii) mentioned above are the first non-trivial computational complexity and algorithmic analyses of this measure of gerrymandering. (b) On the empirical side, we provide a simple and fast algorithm that can “un-gerrymander” the district maps for the states of Texas, Virginia, Wisconsin and Pennsylvania (based on the efficiency gap measure) by bring their efficiency gaps to acceptable levels from the current unacceptable levels. To the best of our knowledge, ours is the first publicly available implementation and its corresponding evaluation on real data for any algorithm for the efficiency gap measure. Our work thus shows that, notwithstanding the general worst-case approximation hardness of the efficiency gap measure as shown by us, finding district maps with acceptable levels of efficiency gaps could be a computationally tractable problem from a practical point of view. Based on these empirical results, we also provide some interesting insights into three practical issues related the efficiency gap measure.

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游击队效率差距度量的理论和经验算法分析

游击队游击队员是美国选民被剥夺选举权的主要原因。但是，迄今为止，说服美国法院采取具体措施量化嫁接数量一直是有限的。最近，斯蒂芬诺普洛斯（Stephanopoulos）和麦基（McGhee）在几篇论文中通过一种所谓的“效率差距”引入了一种新的衡量游击党游荡手段的方法，该效率差距用于计算两党制中两个政党之间浪费的选票的绝对差额。从法律角度看，该措施在法律上令人信服在一家美国上诉法院中，该案声称威斯康星州的立法地图是精美的。本文的目的是形式化并提供最小化该度量的计算问题的理论和经验算法分析。为此，我们显示以下内容：（a）在理论上，我们将相应的最小化问题形式化，并提供最小化效率差距测度问题的非平凡的数学和计算复杂度属性。具体而言，我们针对形式化的最小化问题证明了以下结果：（i）我们表明效率差距测度仅获得有限的离散有理值集。（Cho和Wendy也独立进行了类似性质但使用不同论点的观察（Univ Pa Law Rev 166（1），Article 2，2017）。（ii）我们证明，假设P \（\ ne \ textsf {NP} \），对于一般地图和任意数字选举数据，最小化问题不允许采用近似比率有限的多项式时间算法。而且，我们表明问题仍然存在\（\ textsf {NP} \）-即使数字选举数据在地区数量上是线性的，只要地图以平面图的形式提供（或等效地，在二维欧几里德平面的多边形细分中），也可以完全完成。（iii）尽管有先前的硬度结果，但我们表明，如果对地图和选举数据假设一些合理的限制，则可以设计出有效的精确算法或有效近似算法。上面提到的项目（ii）和（iii）是此重要度量的第一个非平凡的计算复杂度和算法分析。（b）在经验方面，我们提供了一种简单快速的算法，该算法可以“无限制地”处理德克萨斯州，弗吉尼亚州，威斯康星州和宾夕法尼亚州（基于效率差距度量）将其效率差距从当前的不可接受水平提高到可接受水平。据我们所知，我们是第一个可公开获得的实施方案，并且针对任何用于效率差距度量的算法对真实数据进行相应的评估。因此，我们的工作表明，尽管我们展示了效率差距测度的一般最坏情况的近似硬度，但找到具有可接受效率差距水平的地区图可能是从实践的角度讲计算上容易处理的问题。基于这些经验结果，我们还提供了对与效率差距度量相关的三个实际问题的有趣见解。