Algorithms and hardness results for the (k,ℓ)-cover problem
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Bibliographic record
Abstract
A connected graph has a ( k , ℓ ) -cover if each of its edges is contained in at least ℓ cliques of order k . Motivated by recent advances in extremal combinatorics and the literature on edge modification problems, we study the algorithmic version of the ( k , ℓ ) -cover problem. Given a connected graph G , the ( k , ℓ ) -cover problem is to identify the smallest subset of non-edges of G such that their addition to G results in a graph with a ( k , ℓ ) -cover. For every constant k ≥ 3 , we show that the ( k , 1 ) -cover problem is NP -complete for general graphs. Moreover, we show that for every constant k ≥ 3 , the ( k , 1 ) -cover problem admits no polynomial-time constant-factor approximation algorithm unless P = NP . However, we show that the ( 3 , 1 ) -cover problem can be solved in polynomial time when the input graph is chordal. For the class of trees and general values of k , we show that the ( k , 1 ) -cover problem is NP -hard even for spiders. However, we show that for every k ≥ 4 , the ( 3 , k − 2 ) -cover and the ( k , 1 ) -cover problems are constant-factor approximable when the input graph is a tree.
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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