Packing paths in digraphs
Bibliographic record
Abstract
Abstract Let ${\cal G}$ be a fixed set of digraphs. Given a digraph H , a ${\cal G}$ ‐packing in H is a collection ${\cal P}$ of vertex disjoint subgraphs of H , each isomorphic to a member of ${\cal G}$ . A ${\cal G}$ ‐packing ${\cal P}$ is maximum if the number of vertices belonging to members of ${\cal P}$ is maximum, over all ${\cal G}$ ‐packings. The analogous problem for undirected graphs has been extensively studied in the literature. The purpose of this paper is to initiate the study of digraph packing problems. We focus on the case when ${\cal G}$ is a family of directed paths. We show that unless ${\cal G}$ is (essentially) either $\{ \vec {P}_1 \}$ , or $\{ \vec {P}_1, \vec {P}_2 \}$ , the G ‐packing problem is NP‐complete. When ${\cal G} = \{ \vec {P}_1 \}$ , the ${\cal G}$ ‐packing problem is simply the matching problem. We treat in detail the one remaining case, ${\cal G} = \{ \vec {P}_1, \vec {P}_2 \}$ . We give in this case a polynomial algorithm for the packing problem. We also give the following positive results: a Berge type augmenting configuration theorem, a min‐max characterization, and a reduction to bipartite matching. These results apply also to packings by the family ${\cal G}$ consisting of all directed paths and cycles. We also explore weighted variants of the problem and include a polyhedral analysis. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 81–94, 2003
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How this classification was reachedexpand
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 | 0.005 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".