We study an application of fair division theory to school redistricting. Procaccia, Robinson, and Tucker-Foltz (SODA 2024) recently proposed a mathematical model to generate redistricting plans that provide theoretically guaranteed fairness among demographic groups of students. They showed that an almost proportional allocation can be found by adding $O(g \log g)$ extra seats in total, where $g$ is the number of groups. In contrast, for three or more groups, adding $o(n)$ extra seats is not sufficient to obtain an almost envy-free allocation in general, where $n$ is the total number of students. In this paper, we focus on the case of two groups. We introduce a relevant relaxation of envy-freeness, termed 1-relaxed envy-freeness, which limits the capacity violation not in total but at each school to at most one. We show that there always exists a 1-relaxed envy-free allocation, which can be found in polynomial time.

翻译：我们研究了公平分配理论在学区重划中的应用。Procaccia、Robinson 和 Tucker-Foltz（SODA 2024）最近提出了一种数学模型，用于生成在学生群体之间提供理论保证公平性的重划方案。他们证明，通过总共增加 $O(g \log g)$ 个额外名额（其中 $g$ 为群体数量），可以找到一种近似比例分配。相比之下，对于三个或更多群体，通常增加 $o(n)$ 个额外名额不足以获得近似无妒忌的分配，其中 $n$ 为学生总数。在本文中，我们聚焦于两个群体的情况。我们引入了一种无妒忌的相关松弛概念，称为1-松弛无妒忌，它限制每所学校的容量违反量最多为一，而非总量。我们证明，总存在一种1-松弛无妒忌分配，并可在多项式时间内找到。