Good quality virtual realization of unit disk graphs
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We consider the problem of finding a realization of an n-vertex unit disk graph (UDG) expressed in general form, say, as an adjacency matrix. The problem is to construct an embedding of the graph in low-dimensional Euclidean space so that the ratio of the length of the longest edge under the embedding to the length of the shortest non-edge under the embedding is as small as possible; the measure is known as the quality of the realization. Thus, an optimum quality realization has quality between 1/2 and 1. Kuhn et al. gave aO(log3.5 n (loglog n)1/2}) quality realization that requires solving a linear program with exponentially many constraints by using the ellipsoid algorithm. In this article, we give a combinatorial algorithm that achieves an O(log3 n) quality realization of an n-vertex UDG expressed in general form. Thus, not only is our algorithm an improvement, it also bypasses the standard and costly technique of solving a linear program with exponentially many “spreading constraints.” As a side effect of our construction, we get the first constant-factor approximation to the minimum clique partition problem for UDGs expressed in general form. Such a clique partition also represents our key technical contribution. If the embedding is allowed to reside in higher dimensional space, we obtain improved results: a quality-2 embedding in constant dimensional Euclidean space.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.005 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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