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We introduce a new two-sided stable matching problem that describes thesummer internship matching practice of an Australian university. The model is acase between two models of Kamada and Kojima on matchings with distributionalconstraints. We study three solution concepts, the strong and weak stabilityconcepts proposed by Kamada and Kojima, and a new one in between the two,called cutoff stability. Kamada and Kojima showed that a strongly stablematching may not exist in their most restricted model with disjoint regionalquotas. Our first result is that checking its existence is NP-hard. We thenshow that a cutoff stable matching exists not just for the summer internshipproblem but also for the general matching model with arbitrary heredityconstraints. We present an algorithm to compute a cutoff stable matching andshow that it runs in polynomial time in our special case of summer internshipmodel. However, we also show that finding a maximum size cutoff stable matchingis NP-hard, but we provide a Mixed Integer Linear Program formulation for thisoptimisation problem.

Cutoff stability under distributional constraints with an application to summer internship matching