From Matching with Diversity Constraints to Matching with Regional Quotas

TL;DR

This paper establishes a formal connection between two complex matching models with distributional constraints, enabling the transfer of computational and stability results between them.

Contribution

It introduces a polynomial-time reduction from school choice with diversity constraints to hospital-doctor matching with regional quotas, linking their theoretical properties.

Findings

NP-completeness of checking feasible and stable matchings extends to both models.

The reduction preserves feasibility and stability of matchings.

Results in hospital-doctor matching can inform school choice matching.

Abstract

In the past few years, several new matching models have been proposed and studied that take into account complex distributional constraints. Relevant lines of work include (1) school choice with diversity constraints where students have (possibly overlapping) types and (2) hospital-doctor matching where various regional quotas are imposed. In this paper, we present a polynomial-time reduction to transform an instance of (1) to an instance of (2) and we show how the feasibility and stability of corresponding matchings are preserved under the reduction. Our reduction provides a formal connection between two important strands of work on matching with distributional constraints. We then apply the reduction in two ways. Firstly, we show that it is NP-complete to check whether a feasible and stable outcome for (1) exists. Due to our reduction, these NP-completeness results carry over to setting (2). In view of this, we help unify some of the results that have been presented in the literature. Secondly, if we have positive results for (2), then we have corresponding results for (1). One key conclusion of our results is that further developments on axiomatic and algorithmic aspects of hospital-doctor matching with regional quotas will result in corresponding results for school choice with diversity constraints.