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

TL;DR

This paper introduces a new stable matching concept called cutoff stability, applicable to internship matching with distributional constraints, providing algorithms and complexity results for existence and optimization.

Contribution

It defines cutoff stability, proves existence in general models, and offers polynomial algorithms and NP-hardness results for maximum size matchings.

Findings

Cutoff stable matchings exist in general models with heredity constraints.

Existence checking of strong stability is NP-hard.

Polynomial-time algorithm for cutoff stable matchings in the internship model.

Abstract

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