Maximally Diverse Stable Matchings: Optimizing Arbitrary Institutional Objectives

2026-04-30

AI Summary

This work addresses the challenge of optimizing complex institutional objectives—such as multidimensional diversity quotas and sibling co-enrollment—within the constraints of stable matching, a setting where traditional approaches often compromise stability or become computationally intractable. The paper presents the first polynomial-time algorithmic framework capable of efficiently optimizing any computable institutional objective, including utilitarian, egalitarian, or distributional goals defined by general set functions, without sacrificing stability. By leveraging structural properties of stable matchings, the method reformulates the objective as edge weights and integrates graph-theoretic and combinatorial optimization techniques to identify an optimal solution among all stable matchings. The framework is successfully applied to minimize quota violations and maximize sibling co-assignment, demonstrating both theoretical soundness and practical efficacy.

Abstract

Stable matching theory is the foundation of centralized clearinghouses worldwide, from school choice programs to medical residency allocations. However, incorporating complex distributional goals-such as multi-dimensional diversity quotas or sibling co-assignment guarantees-often compromises stability or renders the problem computationally intractable. The existing literature typically addresses this tension by weakening stability to accommodate distributional constraints. In contrast, the reverse question remains largely unexplored: if we restrict attention to stable matchings, to what extent can such distributional objectives be achieved? In this paper, we resolve this tension by introducing a general, polynomial-time algorithmic framework to optimize arbitrary institutional (or even two-sided) objectives within the set of stable matchings. We prove that for any polynomial-time computable set functions $g_i$ evaluating the assigned students at institutions $i \in I$, a stable matching minimizing either the utilitarian objective $\sum_{i\in I} g_i$ or the egalitarian objective $\max_{i\in I} g_i$ can be found efficiently. Our approach leverages the structural properties of stable matchings, mapping arbitrary set functions to linear edge weights. We apply this theorem to efficiently solve major open practical problems: finding stable matchings that minimally violate overlapping diversity quotas (under both total and maximum violations) and maximizing the number of sibling families assigned to the same institution. Even when the distributional objective is prioritized, our algorithm helps to quantify the ``price of stability'', i.e., the gap between the maximally diverse matching and the maximally diverse stable matching.

Problem

• stable matching
• distributional objectives
• diversity quotas
• sibling co-assignment
• institutional objectives

Innovation

• stable matching
• distributional objectives
• polynomial-time algorithm
• diversity quotas
• sibling co-assignment