Searching for optimal integer solutions to set partitioning problems using column generation
Bibliographic record
Abstract
Abstract In this paper, we describe a new approach to increase the possibility of finding integer feasible columns to a set partitioning problem (SPP) directly in solving the linear programming (LP) relaxation using column generation. Traditionally, column generation is aimed to solve the LP‐relaxation as quickly as possible without any concern for the integer properties of the columns formed. In our approach, we aim to generate columns forming an optimal integer solution while simultaneously solving the LP‐relaxation. Using this approach, we can improve the possibility of finding integer solutions by heuristics at each node in the branch‐and‐bound search. In addition, we improve the possibility of finding high‐quality integer solutions in cases where only the columns in the root node are used to solve the problem. The basis of our approach is a subgradient technique applied to a Lagrangian dual formulation of the SPP extended with an additional surrogate constraint. This extra constraint is not relaxed and is used to better control the subgradient evaluations and how the multiplier values are computed. The column generation is then directed, via the multipliers, to construct columns that form feasible integer solutions. Computational experiments show that we can generate optimal integer columns in a large set of well‐known test problems as compared to both standard and stabilized column generation, and simultaneously keep the number of columns smaller than standard column generation. This is also supported by tests on a case study with work‐shift generation.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| 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 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".