A submodular measure and approximate Gomory-Hu theorem for packing odd trails
Bibliographic record
Abstract
Motivated by a problem about totally odd immersions of graphs, we define the odd edge-connectivity λo(u, υ) as the maximum number of edge-disjoint trails of odd length from u to υ. It was recently discovered that λo(u, υ) can be approximated up to a constant multiplicative factor using the usual edge-connectivity between u and v and the minimum value of another parameter that measures “how far from a bipartite graph” the part of the graph around u and v is. In this paper, we formalize this second ingredient and call it the perimeter. We prove that perimeter is a submodular function on the vertex-sets of a graph. Using this fact, we obtain a version of the Gomory–Hu Theorem in which minimum edge-cuts are replaced by sets of minimum perimeter. We construct (in polynomial time) a rooted forest structure, analogous to the Gomory-Hu tree of a graph, which encodes a collection of minimum-perimeter vertex-sets. Although the classical Gomory-Hu Theorem extends to arbitrary symmetric submodular functions, our result is novel and indicates a possibility for further generalizations. These results have significant implications for the study of path and trail systems with parity constraints. We present two such applications: an efficient data structure for storing approximate odd edge-connectivities for all pairs of vertices in a graph, and a rough structure theorem for graphs with no “totally odd” immersion of a large complete graph.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".