Optimization and packings of T-joins and T-cuts
Bibliographic record
Abstract
Let G be a graph and T an even cardinality subset of its vertices. We call (G,T) a graft. A T-join is a subgraph of G whose odd-degree vertices are precisely those in T, and a T-cut is a cut delta(S) where S contains an odd number of vertices of T. An interesting question from a combinatorial optimization perspective is that of finding optimal T-joins and T-cuts. These have applications in various places. We give an overview of several such optimization problems, as well as several algorithms for finding optimal T-joins and T-cuts from the literature.We then consider a packing problem in grafts. It is a simple observation that the number of edge-disjoint T-joins is at most the number of edges in any T-cut. However it is not known exactly when these quantities are equal. It has been conjectured by Guenin that if G is planar and all T-cuts of G have the same parity and the size of every T-cut is at least k, then G contains k edge-disjoint T-joins. The case k = 3 is equivalent to the Four Colour Theorem, and the cases k = 4, which was conjectured by Seymour, and k = 5 were proved by Guenin. Recently, the case k = 6 was settled by Dvorak, Kawarabayashi and Kral. In this thesis, we give a proof of the case k = 7.
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 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".