Numerically Safe Lower Bounds for the Capacitated Vehicle Routing Problem
Bibliographic record
Abstract
The resolution of integer programming problems is typically performed via branch and bound. Nodes of the branch-and-bound tree are pruned whenever the corresponding subproblem is proven not to contain a solution better than the best solution found so far. This is a key ingredient for achieving reasonable solution times. However, since subproblems are solved in floating-point arithmetic, numerical errors can occur and may lead to inappropriate pruning. As a consequence, optimal solutions may be cut off. We propose several methods for avoiding this issue, in the special case of a branch-cut-and-price formulation for the capacitated vehicle routing problem. The methods are based on constructing dual feasible solutions for the linear programming relaxations of the subproblems and obtaining, by weak duality, bounds on their objective function value. Such approaches have been proposed before for formulations with a small number of variables (dual constraints), but the problem becomes more complex when the number of variables is exponentially large, which is the case in consideration. We show that, in practice, along with being safe, our bounds are stronger than those usually employed, obtained with unsafe floating-point arithmetic plus some heuristic tolerance, and all of this at a negligible computational cost. We also discuss some potential advantages and other uses of our safe bounds derivation. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0747 .
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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 | 0.008 | 0.045 |
| Meta-epidemiology (narrow) | 0.004 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.005 | 0.006 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.002 | 0.009 |
| Insufficient payload (model declined to judge) | 0.012 | 0.004 |
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".