Trimming Forests Is Hard (Unless They Are Made of Stars)
Bibliographic record
Abstract
Abstract. Graph modification problems ask for the minimal number of vertex/edge additions/deletions needed to make a graph satisfy some predetermined property. A (meta-)problem of this type, which was raised by Yannakakis in 1981, asks to determine for which properties [Formula: see text] it is NP-hard to compute the smallest number of edge deletions needed to make a graph satisfy [Formula: see text]. Despite being extensively studied in the past 40 years, this problem is still wide open. In fact, it is open even when [Formula: see text] is the property of being [Formula: see text]-free, for some fixed graph [Formula: see text]. In this case we use [Formula: see text] to denote the smallest number of edge deletions needed to turn [Formula: see text] into an [Formula: see text]-free graph. Alon, Shapira, and Sudakov proved that if [Formula: see text] is not bipartite, then computing [Formula: see text] is NP-hard. They left open the problem of classifying the bipartite graphs [Formula: see text] for which computing [Formula: see text] is NP-hard. In this paper we resolve this problem when [Formula: see text] is a forest, showing that computing [Formula: see text] is polynomial-time solvable if [Formula: see text] is a star forest and NP-hard otherwise. Our main innovation in this work lies in introducing a new graph-theoretic approach for Yannakakis’s problem, which differs significantly from all prior works on this subject. In particular, we prove new results concerning an old and famous conjecture of Erdős and Sós, which are of independent interest.
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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 | 0.001 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.004 | 0.011 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.003 | 0.004 |
| Insufficient payload (model declined to judge) | 0.016 | 0.004 |
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".