Bibliographic record
Abstract
A disk graph is an intersection graph of disks in $\mathbb{R}^2$. Determining the computational complexity of finding a maximum clique in a disk graph is a long-standing open problem. In 1990, Clark, Colbourn, and Johnson gave a polynomial-time algorithm for computing a maximum clique in a unit disk graph. However, finding a maximum clique when disks are of arbitrary size is widely believed to be a challenging open problem. The problem is open even if we restrict the disks to have at most two different sizes of radii, or restrict the radii to be within $[1,1+\varepsilon]$ for some $ε>0$. In this paper, we provide a new perspective to examine adjacencies in a disk graph that helps obtain the following results. - We design an $O(2^k n^{2k} poly(n))$-time algorithm to find a maximum clique in a $n$-vertex disk graph with $k$ different sizes of radii. This is polynomial for every fixed $k$, and thus settles the open question for the case when $k=2$. - Given a set of $n$ unit disks, we show how to compute a maximum clique inside each possible axis-aligned rectangle determined by the disk centers in $O(n^5\log n)$-time. This is at least a factor of $n^{4/3}$ faster than applying the fastest known algorithm for finding a maximum clique in a unit disk graph for each rectangle independently. - We give an $O(2^kn^{2rk} poly(n,r))$-time algorithm to find a maximum clique in a $n$-vertex ball graph with $k$ different sizes of radii where the ball centers lie on $r$ parallel planes. This is polynomial for every fixed $k$ and $r$, and thus contrasts the previously known NP-hardness result for finding a maximum clique in an arbitrary ball graph.
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.003 | 0.001 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.011 | 0.002 |
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".