Valid Inequalities and a Branch-and-Cut Algorithm for Asymmetric Multi-Depot Routing Problems
Bibliographic record
Abstract
We present a generic branch-and-cut framework for solving routing problems with multiple depots and asymmetric cost-structures, which consist in finding a set of cost minimizing (capacitated) vehicle tours in order to fulfill a set of customer demands. The backbone of the branch-and-cut framework is a series of valid inequalities, and accompanying separation algorithms, exploiting the asymmetric cost-structure in directed graphs. We derive three new classes of so-called DK inequalities that can eliminate subtours, enforce tours to be linked to a single depot, and impose bounds on the number of allowed customers in a tour. In addition, other well-known valid inequalities for solving vehicle routing problems are generalized and adapted to be valid for routing problems with multiple depots and asymmetric cost-structures. The resulting branch-and-cut framework is tested on four specific problem variants, for which we develop a new set of large-scale benchmark instances. The new DK inequalities are able to reduce root node optimality gaps by up to 67.2% compared to existing approaches in the literature. The overall branch-and-cut framework is effective as, e.g., Asymmetric Multi-Depot Traveling Salesman Problem instances containing up to 400 customers and 50 depots can be solved to optimality, for which only solutions of instances up to 300 customer nodes and 60 depots were reported in the literature before.
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| 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".