Compatible $k$-Relaxations of Fairness and Non-Wastefulness Under Hereditary Constraints

2026-04-30

AI Summary

This work addresses the inherent tension between fairness and non-wastefulness in two-sided matching markets under hereditary constraints, where these two properties are typically incompatible. The authors propose a symmetric and controllable relaxation of both criteria, formalized as ER-$k$ (for fairness) and NW-$k$ (for non-wastefulness), and prove that for any fixed $k$, the relaxed properties are always simultaneously achievable—thereby transcending the conventional trade-off paradigm. Building on this insight, they design two equivalent polynomial-time algorithms: the $k$-acceptable cutoff algorithm and the $k$-acceptable college-proposing deferred acceptance mechanism. Empirical results demonstrate that even minimal values of $k$ suffice to achieve a strong balance between fairness and non-wastefulness in practice.

Abstract

We study two-sided matching markets under hereditary constraints, which extend beyond simple capacity limits and arise in applications such as diversity requirements and refugee resettlement. In these settings, fairness and non-wastefulness are often incompatible, and existing approaches typically address this tension by prioritizing one property at the expense of the other. We take a different approach by relaxing both properties simultaneously in a controlled and symmetric manner. We introduce two notions indexed by an integer $k$: envy-received up to $k$ peers (ER-$k$) and non-wastefulness up to $k$ objections (NW-$k$). Our main theoretical result shows that ER-$k$ and NW-$k$ are always compatible under hereditary constraints for any fixed $k$. We provide two equivalent polynomial-time algorithms to compute such matchings: a $k$-admissible cutoff algorithm and a $k$-admissible college-proposing deferred acceptance mechanism. Finally, experimental results demonstrate that even small relaxations achieve a favorable balance between fairness and non-wastefulness.

Problem

• fairness
• non-wastefulness
• hereditary constraints
• two-sided matching
• compatibility

Innovation

• hereditary constraints
• k-relaxations
• envy-received up to k peers
• non-wastefulness up to k objections
• deferred acceptance mechanism