Approximation algorithms for maximum weighted internal spanning trees in regular graphs and subdivisions of graphs
Bibliographic record
Abstract
Abstract Let $G$ be a vertex-weighted connected graph of $n$ vertices and let $T$ be a spanning tree of $G$. We call $T$ a maximum weighted internal spanning tree of $G$ if the sum of the weights of the internal vertices of $T$ is the maximum over all spanning trees of $G$. The maximum weighted internal spanning tree (MaxwIST) problem asks to find such a spanning tree $T$ of $G$. The problem is NP-hard. We give an $O(dn)$ time approximation algorithm for $d$-regular graphs of $n=|V|$ vertices that computes a spanning tree with total weight of the internal vertices is at least $\frac{\beta _{d}}{\beta _{d} +d-2} - \epsilon $ of the total weight of all the vertices of the graph for any $\epsilon>0$, where $\beta _{d} = (d-1)H_{d-1}$, and $H_{d-1} = \sum _{i=1}^{d-1} i^{-1}$ is the $(d-1)$th harmonic number. For every $d \geq 3$ and $n_{0} \geq 1$, we show the construction of a $d$-regular graph of at least $n_{0}$ vertices, such that for any of its spanning trees, $\frac{w(I)}{w(V)}\le \frac{d}{d+1}$ holds. We give an $O(dn)$ time approximation algorithm for subdivisions of $d$-regular graphs, where the ratio of the internal weight of the spanning tree with the total vertex weight of the graph is at least $\frac{d-1}{2d-3} - \epsilon $ for $\epsilon>0$. We extend our study to $x$-subdivisions of Hamiltonian and hypoHamiltonian graphs, where each edge of the original Hamiltonian or hypoHamiltonian graph has been subdivided at least $x$ times. For those two graph classes, we show that there exists a spanning tree with internal vertex weight at least $1-\frac{2}{x-1}$ of the total vertex weight of the graph. Furthermore, we give $O(n)$ time algorithm for $x$-subdivisions of biconnected outerplanar graphs and $4$-connected planar graphs to achieve the above bound.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".