Capacity Planning in Stable Matching: An Application to School Choice

In this work, we introduce the problem of jointly allocating a school capacity expansion (given a fixed budget) and finding the best allocation for the students in the expanded market. Given the computational complexity of the problem, we provide an integer quadratically-constrained programming formulation and study its linear reformulations. We also propose two heuristics: A greedy algorithm and an LP-based method. We empirically evaluate the performance of our approaches in a detailed computational study. We observe the practical superiority of the linearized model in comparison with its quadratic counterpart and we outline their computational limits. Finally, we use the Chilean school choice system data to empirically demonstrate the impact of capacity planning under stability conditions. Our results show that each additional school seat can benefit multiple students. In addition, depending on the decision-maker, our methodology can prioritize the assignment of previously unassigned students or improve the assignment of several students through improvement chains.