Envy-Free School Redistricting Between Two Groups

TL;DR

This paper explores fair school redistricting between two groups, introducing a relaxed envy-freeness concept that guarantees a fair allocation with limited capacity violations, and provides a polynomial-time algorithm for finding such allocations.

Contribution

It introduces 1-relaxed envy-freeness for two-group redistricting and proves the existence and polynomial-time computability of such allocations.

Findings

Existence of 1-relaxed envy-free allocations for two groups.

Polynomial-time algorithm for finding these allocations.

Limited capacity violations at each school.

Abstract

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.