A linear-time algorithm to find a separator in a graph excluding a minor
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Bibliographic record
Abstract
Let G be an n -vertex m -edge graph with weighted vertices. A pair of vertex sets A , B ⊆ V ( G ) is a 2/3 -separation of order | A ∩ B | if A ∪ B = V ( G ), there is no edge between A − B and B − A , and both A − B and B − A have weight at most 2/3 the total weight of G . Let ℓ ∈ Z + be fixed. Alon et al. [1990] presented an algorithm that in O ( n 1/2 m ) time, outputs either a K ℓ -minor of G , or a separation of G of order O ( n 1/2 ). Whether there is a O ( n + m )-time algorithm for this theorem was left as an open problem. In this article, we obtain a O ( n + m )-time algorithm at the expense of a O ( n 2/3 ) separator. Moreover, our algorithm exhibits a trade-off between time complexity and the order of the separator. In particular, for any given ϵ ∈ [0,1/2], our algorithm outputs either a K ℓ -minor of G , or a separation of G with order O ( n (2−ϵ)/3 in O ( n 1 + ϵ + m ) time. As an application we give a fast approximation algorithm for finding an independent set in a graph with no K ℓ-minor.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 it