We investigate the following many-to-one stable matching problem with diversity constraints (SMTI-Diverse): Given a set of students and a set of colleges which have preferences over each other, where the students have overlapping types, and the colleges each have a total capacity as well as quotas for individual types (the diversity constraints), is there a matching satisfying all diversity constraints such that no unmatched student-college pair has an incentive to deviate?
Here, an unmatched student-college pair has an *incentive* to deviate if matching it (possibly by unmatching some other pairs) results in a strictly better solution for both the student and the college in the pair.
SMTI-Diverse is known to be NP-hard. However, as opposed to the NP-membership claims in the literature (Aziz et al., AAMAS 2019; Huang, SODA 2010), we prove that it is beyond NP: it is complete for the complexity class .

In addition, we provide a comprehensive analysis of the problem’s complexity from the viewpoint of natural restrictions to inputs such as bounding the number of colleges , students , types , and/or the maximum upper quota , and the maximum capacity , and obtain new algorithms for the problem.

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The conference version and the long version of the paper can be found at IJCAI20 and on arXiv, respectively.

For more information on the complexity classes NP and Σ^{P}_{2}, please refer to e.g., Wikipedia NP and Polynomial hierarchy