Canonical cuts of path powers
Bibliographic record
Abstract
The MaxCut problem aims to find a bipartition of vertices in a given graph such that the number of edges with one end vertex in each part is maximum among all bipartitions. NP-hardness when restricted to interval graphs has been recently announced. Surprisingly, all previously published attempts at polynomial-time algorithms for unit interval graphs turned out to be wrong, which justifies the search for subclasses where MaxCut can be handled. We introduce canonical cuts whose pattern allows an easy computation of the cut size for the power of paths $P_n^k$. Using canonical cuts, we calculate the structure and the size of maximum cuts for $k\leq 5$ and for $n\leq \frac{2}{3}(2k+2)$. We prove that the known size for a maximum cut for reduced co-bipartite chain graphs can be achieved by a canonical cut. We perform computational experiments on each $P_n^k$ graph with $1\leq k\leq n\leq 43$ and show that most of them allow a canonical cut that is maximum. We display a table with the found cases where there is no canonical cut which is a maximum cut. In these graphs, the difference between the maximum cut and some canonical is at most 3 units. This indicates canonical cuts as a good approach to tackle the maximum cut on $P_n^k$ graphs.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".