To handle various applications, we study matching under constraints. The only requirement on the constraints is heredity; given a feasible matching, any matching with fewer students at each school is also feasible. Heredity subsumes existing constraints such as regional maximum quotas and diversity constraints. With constraints, there may not exist a matching that satisfies fairness and nonwastefulness (i.e., stability). We demonstrate our new mechanism, the Adaptive Deferred Acceptance mechanism (ADA), satisfies strategy-proofness for students, nonwastefulness, and a weaker fairness property. We also offer a technique to apply ADA even if heredity is violated (e.g., minimum quotas).
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