We model the societal task of redistricting political districts as a partitioning problem: Given a set of $n$ points in the plane, each belonging to one of two parties, and a parameter $k$, our goal is to compute a partition $\Pi$ of the plane into regions so that each region contains roughly $\sigma = n/k$ points. $\Pi$ should satisfy a notion of ''local'' fairness, which is related to the notion of core, a well-studied concept in cooperative game theory. A region is associated with the majority party in that region, and a point is unhappy in $\Pi$ if it belongs to the minority party. A group $D$ of roughly $\sigma$ contiguous points is called a deviating group with respect to $\Pi$ if majority of points in $D$ are unhappy in $\Pi$. The partition $\Pi$ is locally fair if there is no deviating group with respect to $\Pi$. This paper focuses on a restricted case when points lie in $1$D. The problem is non-trivial even in this case. We consider both adversarial and ''beyond worst-case" settings for this problem. For the former, we characterize the input parameters for which a locally fair partition always exists; we also show that a locally fair partition may not exist for certain parameters. We then consider input models where there are ''runs'' of red and blue points. For such clustered inputs, we show that a locally fair partition may not exist for certain values of $\sigma$, but an approximate locally fair partition exists if we allow some regions to have smaller sizes. We finally present a polynomial-time algorithm for computing a locally fair partition if one exists.
翻译:我们把重新划分政治区的社会任务作为一个分割问题来模拟:鉴于飞机上一组美元点,每个属于两个政党之一,每个属于一个政党,一个参数为美元,我们的目标是将飞机的一个分区$\Pi美元计算成区域,以便每个区域都包含大约$\gma=n/k美元点。$\Pi美元应该满足“当地”公平的概念,这与核心概念有关,这是合作游戏理论中一个经过深思熟虑的概念。一个区域与该区域的多数政党有关,如果属于少数政党,则美元是一个不满意的点。一个大约为美元\Pi$的分组将连结点计算成一个偏差,如果以美元计的多数的点不满意,则每个区域应该满足“当地”的公平概念。如果不存在与美元有关的反差组,那么这个区域就是一个有限的案例,如果我们以1美元计价的点为1美元,那么,我们就会看到一个最差的本地的计算值,那么问题就是一个非当地货币的计算值。 问题就是一个本地的货币的计算。如果存在一种最差的货币的计算法的计算,那么,那么,我们就会看到一个地方的计算,一个地方的计算,一个地方的计算,一个地方的计算就是一个地方的计算。一个地方的货币的计算法的计算, 问题就是一个地方的计算。我们就会有一个地方的计算。一个不偏差的计算。