Envy-Free School Redistricting Between Two Groups

2026-03-20

AI Summary

This work addresses the school redistricting problem involving two student groups by proposing a 1-relaxed envy-free allocation scheme: under the local constraint that each school’s capacity may exceed its limit by at most one student, envy-freeness between the two groups is guaranteed. By introducing the notion of 1-relaxed envy-freeness, the traditional global capacity constraint is transformed into school-level local constraints. Combining concepts from fair division theory with combinatorial optimization techniques, the authors design a polynomial-time algorithm and prove that, for any instance with two groups, a fair allocation satisfying this relaxed condition always exists and can be efficiently constructed—thereby achieving both fairness and computational tractability.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• envy-free
• school redistricting
• fair division
• two groups
• capacity violation

Innovation

• envy-freeness
• school redistricting
• fair division
• polynomial-time algorithm
• capacity relaxation