Maximally Diverse Stable Matchings: Optimizing Arbitrary Institutional Objectives

TL;DR

This paper introduces a polynomial-time algorithmic framework to optimize diverse institutional objectives within the set of stable matchings, balancing stability with complex distributional goals.

Contribution

It provides a novel method to efficiently find stable matchings that optimize arbitrary distributional objectives, such as diversity quotas and sibling co-assignment.

Findings

Efficiently finds stable matchings minimizing violations of diversity quotas.

Maximizes sibling family assignments within stable matchings.

Quantifies the trade-off between diversity objectives and stability.

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.