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Using school choice as a motivating example, we introduce a stylized model of a many-to-one matching market where the clearinghouse aims to implement contingent priorities, i.e., priorities that depend on the current assignment, to prioritize students with siblings and match them together. We provide a series of guidelines and introduce two natural approaches to implement them: (i) absolute, whereby a prioritized student can displace any student without siblings assigned to the school, and (ii) partial, whereby prioritized students can only displace students that have a less favorable lottery than their priority provider. We study several properties of the corresponding mechanisms, including the existence of a stable assignment under contingent priorities, the complexity of deciding whether there exists one, and its incentive properties. Furthermore, we introduce a soft version of these priorities to guarantee existence, and we provide mathematical programming formulations to find such stable matching or certify that one does not exist. Finally, using data from the Chilean school choice system, we show that our framework can significantly increase the number of students assigned to their top preference and the number of siblings assigned together relative to current practice.

Motivated by the shortage of seats that the Chilean school choice system is facing, we introduce the problem of jointly increasing school capacities and finding a student-optimal assignment in the expanded market. Due to the theoretical and practical complexity of the problem, we provide a comprehensive set of tools to solve the problem, including different mathematical programming formulations, a cutting plane algorithm, and two heuristics that allow obtaining near-optimal solutions quickly. On the theoretical side, we show the correctness of our formulations, different properties of the objective and feasible region that facilitate computation, and also several properties of the underlying mechanism to find a student-optimal matching under capacity expansions. On the computational side, we use data from the Chilean school choice system to demonstrate the impact of our framework and derive insights that could help alleviate the problem. Our results show that each additional seat can benefit multiple students and that we can effectively target the assignment of previously unassigned students or improve the assignment of several students through improvement chains. Nevertheless, our results show that the marginal effect of each additional seat is decreasing and that simply adding seats is insufficient to ensure every student gets assigned to some school. Finally, we discuss several extensions of our framework, showcasing its flexibility to accommodate different needs.