Approximation algorithms for non-sequential star packing problems
Bibliographic record
Abstract
For a positive integer k ≥ 1 , a k -star ( k + -star, k − -star, respectively) is a connected graph containing a degree- ℓ vertex and ℓ degree-1 vertices, where ℓ = k ( ℓ ≥ k , 1 ≤ ℓ ≤ k , respectively). The k + -star packing problem is to cover as many vertices of an input graph G as possible using vertex-disjoint k + -stars in G ; and given k > t ≥ 1 , the k − / t -star packing problem is to cover as many vertices of G as possible using vertex-disjoint k − -stars but no t -stars in G . Both problems are NP-hard for any fixed k ≥ 2 . We present a ( 1 + k 2 2 k + 1 ) - and a 3 2 -approximation algorithms for the k + -star packing problem when k ≥ 3 and k = 2 , respectively, and a ( 1 + 1 t + 1 + 1 / k ) -approximation algorithm for the k − / t -star packing problem when k > t ≥ 2 . They are all local search algorithms and they improve the best known approximation algorithms for the problems, respectively.
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".