Towards Tight Bounds on Theta-Graphs
Bibliographic record
Abstract
We present improved upper and lower bounds on the spanning ratio of $θ$-graphs with at least six cones. Given a set of points in the plane, a $θ$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ=2π/m$, and adds an edge to the `closest' vertex in each cone. We show that for any integer $k \geq 1$, $θ$-graphs with $4k+2$ cones have a spanning ratio of $1+2\sin(θ/2)$ and we provide a matching lower bound, showing that this spanning ratio tight. Next, we show that for any integer $k \geq 1$, $θ$-graphs with $4k+4$ cones have spanning ratio at most $1+2\sin(θ/2)/(\cos(θ/2)-\sin(θ/2))$. We also show that $θ$-graphs with $4k+3$ and $4k+5$ cones have spanning ratio at most $\cos(θ/4)/(\cos(θ/2)-\sin(3θ/4))$. This is a significant improvement on all families of $θ$-graphs for which exact bounds are not known. For example, the spanning ratio of the $θ$-graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. These spanning proofs also imply improved upper bounds on the competitiveness of the $θ$-routing algorithm. In particular, we show that the $θ$-routing algorithm is $(1+2\sin(θ/2)/(\cos(θ/2)-\sin(θ/2)))$-competitive on $θ$-graphs with $4k+4$ cones and that this ratio is tight. Finally, we present improved lower bounds on the spanning ratio of these graphs. Using these bounds, we provide a partial order on these families of $θ$-graphs. In particular, we show that $θ$-graphs with $4k+4$ cones have spanning ratio at least $1+2\tan(θ/2)+2\tan^2(θ/2)$. This is somewhat surprising since, for equal values of $k$, the spanning ratio of $θ$-graphs with $4k+4$ cones is greater than that of $θ$-graphs with $4k+2$ cones, showing that increasing the number of cones can make the spanning ratio worse.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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 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".