Weak duality for packing edge-disjoint odd (u, v)-trails
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Bibliographic record
Abstract
Despite Menger's famous duality between packings and coverings of (u, v)-paths in a graph, there is no duality when we require the paths be odd: a graph with no two edge-disjoint (u, v)-paths may need an arbitrarily large number of edges to cover all such paths. In this paper, we study the relaxed problem of packing trails. Our main result is an approximate duality for trails: if v(u, v) denotes the maximum number of edge-disjoint (u, v)-trails of in a graph G and t (u, v) denotes the minimum number of edges that intersect every such trail, then[EQUATION]The proof leads to a polynomial-time algorithm to find, for any given k, either k edge-disjoint (u, v)-trails or a set of fewer than 8k edges intersecting all (u, v)-trails. This yields a constant factor approximation algorithm for the packing number v(u, v).This result generalizes to the setting of signed graphs and to the setting of group-labelled graphs, in which case odd length is replaced by non-unit product of labels. The motivation for this result comes from the study of totally graph immersions, and our results explain, in particular, why there is an essential difference between the totally weak and strong immersions.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it