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

TL;DR

This paper introduces a framework for relaxing fairness and non-wastefulness in two-sided matching markets with hereditary constraints, ensuring compatibility and efficient computation.

Contribution

It proposes two new notions, ER-$k$ and NW-$k$, that allow controlled relaxation of fairness and non-wastefulness, with polynomial algorithms ensuring their compatibility.

Findings

ER-$k$ and NW-$k$ are always compatible under hereditary constraints for any fixed $k$

Two polynomial-time algorithms are provided for computing such matchings

Small relaxations can effectively balance fairness and non-wastefulness in experiments

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.