Tilted inequalities and facets of the set covering polytope: A theoretical analysis
Bibliographic record
Abstract
Given a ground-set of elements and a family of subsets, the set covering problem consists in choosing a minimum number of elements such that each subset contains at least one of the chosen elements. This research focuses on the set covering polytope , which is the convex hull of integer solutions to the set covering problem. We investigate the connection between the study of the facets of the set covering polytope and tilting theory. This theory studies how inequalities can be rotated around their contact points with a polyhedron in order to obtain inequalities inducing higher dimensional faces . To study this connection, we introduce the concept of tilting vectors which characterize the degrees of freedom of rotation of an inequality. These vectors characterize facet-defining inequalities and can be used to tilt inequalities with a similar procedure to the one used for arbitrary polyhedra. Additionally, we demonstrate that the computational effort needed to tilt an inequality can be reduced when the inequality has many null coefficients. Finally, we use the tilting vectors to extend several necessary and/or sufficient conditions for facets of the set covering polytope presented by several previous works of the literature.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.010 | 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".