Neural Sequence Generation with Constraints via Beam Search with Cuts: A Case Study on VRP
Bibliographic record
Abstract
In recent years, neural sequence models have been applied successfully to solve combinatorial optimization problems. Solutions, encoded as sequences, are typically generated from trained models via beam search, a search algorithm that generates sequences token-by-token while keeping a fixed number of promising partial solutions at each step. In this paper, we explore the problem of augmenting beam search generation with the enforcement of requirements---hard constraints that any generated solution must adhere to. We propose a hybrid approach, by encoding the requirements in the form of a constraint satisfaction problem (CSP) and iteratively solving the CSP to cut any partial solution within the beam search that is incapable of satisfying the requirements. We study this problem in the context of vehicle routing problems (VRP) further augmented with capacity-related or temporal requirements. We experimentally show that cuts often allow us to satisfy the requirements with negligible impact on solution quality. Without the use of cuts, beam search is shown to be exponentially less likely to satisfy the requirements as the length of the solution increases and/or the requirements are strengthened.
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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 | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| 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".