A Strong Class of Lifted Valid Inequalities for the Shortest Path Problem in Digraphs with Negative Cost Cycles
Bibliographic record
Abstract
In this paper, we address a strong class of lifted valid inequalities for the shortest path problem in digraphs with possibly negative cost cycles. We call these lifted inequalities the $incident \ lifted \ valid \ inequalities$ ($ILI$) as they are based on the incident arcs of a given vertex. The $ILI$ inequalities are close in spirit of the so-called \textit{simple lifted valid inequalities} ($SLI$) and $cocycle \ lifted \ valid \ inequalities$ ($CLI$) introduced in Ibrahim et al. (2015). However, as we will see the $ILI$ inequalities are stronger than the first ones in term of linear relaxation strengthening. Indeed, contrary to $SLI$ and $CLI$ inequalities, consider the same instances, in a cutting plane algorithm, the computational results prove that the $ILI$ inequalities provide the optimal integer solution for all the considered instances within no more than three iterations except one case for which after the first strengthening iteration, there exists no generated inequality.
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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 | 0.012 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.003 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.007 |
| Insufficient payload (model declined to judge) | 0.008 | 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".