Capacity Expansion in the College Admission Problem.
Bibliographic record
Abstract
The college admission problem plays a fundamental role in several real-world allocation mechanisms such as school choice and supply chain stability. The classical framework assumes that the capacity of each college is known and fixed in advance. However, increasing the quota of even a single college would improve the overall cost of the students. In this work, we study the problem of finding the college capacity expansion that achieves the best cost of the students, subject to a cardinality constraint. First, we show that this problem is NP-hard to solve, even under complete and strict preference lists. We provide an integer quadratically constrained programming formulation and study its linear reformulation. We also propose two natural 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. In terms of solution quality, we note that the allocation of a few extra spots can significantly impact the overall student satisfaction.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".