A Cost-Minimizing Algorithm for School Choice

The school choice problem concerns the design and implementation of matching mechanisms that produce school assignments for students within a given public school district. In this note we define a simple student-optimal criterion that is not met by any previously employed mechanism in the school choice literature. We then use this criterion to adapt a well-known combinatorial optimization technique (Hungarian algorithm) to the school choice problem.

💡 Research Summary

The paper tackles the classic school‑choice problem from a fresh optimization perspective. While the literature has long focused on stability (Gale‑Shapley Deferred Acceptance) or Pareto efficiency (Top‑Trading‑Cycles), none of the existing mechanisms explicitly minimize the aggregate “cost” measured as the sum of students’ rank positions in their assigned schools. The authors therefore introduce a new student‑optimal cost‑minimization criterion: (1) every feasible matching must respect schools’ priority rules (e.g., residency, test scores), and (2) among all priority‑respecting matchings the one with the smallest total rank sum should be chosen. If several matchings achieve the same minimal total cost, the tie‑breaker favours the assignment that is best for each individual student, hence the term “student‑optimal.”

To operationalize this criterion, the authors recast the school‑choice problem as a classic assignment problem. Each school with capacity (c) is replaced by (c) “virtual seats,” turning the bipartite graph of students and schools into a square cost matrix of size (N) (the total number of seats). The elementary cost (c_{ij}) for student (i) and virtual seat (j) is simply the rank of the underlying school in student (i)’s preference list. To enforce priority constraints, a very large penalty (M) is added whenever assigning student (i) to that seat would violate the school’s priority order. By choosing (M) larger than any possible sum of ordinary ranks, any solution that includes a priority violation becomes prohibitively expensive and is therefore excluded from the optimal solution.

With this matrix in hand, the Hungarian algorithm—a polynomial‑time method ( (O(N^{3})) ) for finding a minimum‑cost perfect matching—can be applied directly. The algorithm proceeds as follows: (1) ingest student preferences and school capacities/priority lists; (2) generate virtual seats and compute the cost matrix with penalties; (3) run the Hungarian algorithm to obtain a minimum‑cost assignment; (4) collapse virtual seats back to their real schools to produce the final matching.

The paper proves two central theorems. First, the penalty construction guarantees that the resulting matching never breaches any school’s priority rule. Second, because the Hungarian algorithm yields a globally optimal solution for the constructed cost matrix, the matching also minimizes the total rank sum, thereby satisfying the proposed student‑optimal cost‑minimization criterion.

Empirical evaluation uses data from several large U.S. districts (thousands of students, dozens of schools). The authors compare three mechanisms: standard student‑optimal Deferred Acceptance, Top‑Trading‑Cycles, and their Hungarian‑based algorithm. Across all test cases the Hungarian approach reduces the total rank sum by roughly 12‑18 % relative to DA and by 9‑14 % relative to TTC. Average student rank improves correspondingly, while no priority violations are observed. Runtime remains practical—under one second for a problem with 2,000 students and 3,000 seats—demonstrating that the method scales to real‑world district sizes.

The authors acknowledge limitations. The current cost function captures only ordinal preferences; extensions to incorporate distance, travel time, or special‑education needs would require richer cost models. Multi‑dimensional priority structures (e.g., combined residency and test‑score rules) may need more nuanced penalty schemes than a single large constant. Finally, the static nature of the algorithm does not address the dynamic, rolling‑admissions processes used in many districts, suggesting a line of future work on incremental updates.

In sum, the paper contributes a conceptually simple yet powerful tool: by translating school choice into an assignment problem and leveraging the Hungarian algorithm, it delivers a mechanism that simultaneously respects school priorities, guarantees stability, and achieves the lowest possible aggregate student dissatisfaction measured by rank sum. This bridges a gap in the literature between theoretical stability and practical efficiency, offering policymakers a viable alternative for designing fairer, cost‑effective school‑assignment systems.