Dynamic Aggregation of Set-Partitioning Constraints in Column Generation
Bibliographic record
Abstract
Column generation is often used to solve problems involving set-partitioning constraints, such as vehicle-routing and crew-scheduling problems. When these constraints are in large numbers and the columns have on average more than 8–12 nonzero elements, column generation often becomes inefficient because solving the master problem requires very long solution times at each iteration due to high degeneracy. To overcome this difficulty, we introduce a dynamic constraint aggregation method that reduces the number of set-partitioning constraints in the master problem by aggregating some of them according to an equivalence relation. To guarantee optimality, this equivalence relation is updated dynamically throughout the solution process. Tests on the linear relaxation of the simultaneous vehicle and crew-scheduling problem in urban mass transit show that this method significantly reduces the size of the master problem, degeneracy, and solution times, especially for larger problems. In fact, for an instance involving 1,600 set-partitioning constraints, the master problem solution time is reduced by a factor of 8.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".