  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.
