Integral Column Generation for Set Partitioning Problems with Side Constraints
Bibliographic record
Abstract
The integral column generation algorithm (ICG) was recently introduced to solve set partitioning problems involving a very large number of variables. This primal algorithm generates a sequence of integer solutions with decreasing costs, leading to an optimal or near-optimal solution. ICG combines the well-known column generation algorithm and a primal algorithm called the integral simplex using decomposition algorithm (ISUD). In this paper, we develop a generalized version of ICG, denoted I 2 CG, that can solve efficiently large-scale set partitioning problems with side constraints. This new algorithm can handle the side constraints in the reduced problem of ISUD, in its complementary problem, or in both components. Computational experiments on instances of the airline crew pairing problem (CPP) and the multidepot vehicle routing problem with time windows show that the latter strategy is the most efficient one and I 2 CG significantly outperforms basic variants of two popular column generation heuristics, namely, a restricted master heuristic and a diving heuristic. For the largest tested CPP instance with 1,761 constraints, I 2 CG can produce in less than one hour of computational time more than 500 integer solutions leading to an optimal or near-optimal solution. Summary of Contribution: In this paper, we develop a new integral column generation algorithm that can solve efficiently large-scale set partitioning problems with side constraints. The latter alter the quasi-integrality property needed for primal integral algorithms. The paper adds a methodological contribution remedying this issue. This remedy should, in our opinion, boost the use of primal exact methods, especially in the column generation context. The paper also has a computational contribution. Effectively, computational experiments on instances of the airline crew pairing problem and the multidepot vehicle routing problem with time windows are extensively discussed. We compare the proposed algorithm to basic variants of two popular column generation heuristics.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| 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".