Minimal Envy Matchings in the Hospitals/Residents Problem with Lower Quotas

TL;DR

This paper studies the problem of finding matchings in the Hospitals/Residents setting with both lower and upper quotas that minimize envy, revealing computational hardness and providing an exponential-time solution.

Contribution

It introduces the problem of minimal envy matchings with lower quotas, proves NP-hardness, and offers an exponential-time algorithm for the problem.

Findings

Minimizing envy in HRLQ is NP-hard.

An exponential-time algorithm for minimum-envy-pair problem is proposed.

Feasible matchings satisfying quotas and minimal envy are characterized.

Abstract

In the Hospitals/Residents problem, every hospital has an upper quota that limits the number of residents assigned to it. While, in some applications, each hospital also has a lower quota for the number of residents it receives. In this setting, a stable matching may not exist. Envy-freeness is introduced as a relaxation of stability that allows blocking pairs involving a resident and an empty position of a hospital. While, envy-free matching might not exist either when lower quotas are introduced. We consider the problem of finding a feasible matching that satisfies lower quotas and upper quotas and minimizes envy in terms of envy-pairs and envy-residents in the Hospitals/Resident problem with Lower Quota. We show that the problem is NP-hard with both envy measurement. We also give a simple exponential-time algorithm for the Minimum-Envy-Pair HRLQ problem.