From Matching with Diversity Constraints to Matching with Regional Quotas
This work unifies results for researchers in matching theory, but it is incremental as it builds on existing models without introducing new algorithms or broad SOTA improvements.
The paper tackles the problem of connecting two matching models with distributional constraints by providing a polynomial-time reduction from school choice with diversity constraints to hospital-doctor matching with regional quotas, showing that feasibility and stability are preserved and proving NP-completeness for checking feasible and stable outcomes in both settings.
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.