Multi-guard covers for polygonal regions
Bibliographic record
Abstract
We study the problem of nding optimal covers of polygonal regions using multiple mobile guards. By our denition, a point is covered if, at some time, it lies within the convex hull of the guards from which it is visible. The denition captures our desire that guards both \see and \surround points that they cover. Guards move along continuous timeparameterized curves within a polygonal region P . An optimal m-guard cover of P is a set of m guard paths of minimum total length that cover all points in P . In this paper, we restrict our attention to the case where P is convex, and m is either two or three. We rst address the apparently simpler problem of optimally covering all points on the boundary, @P , of P . Although the guard paths are not restricted to @P , we prove that in every optimal two-guard boundary cover the guards remain on @P . When there are three guards, an optimal boundary cover may require a guard to cross the interior of the polygon. We show, however, that every optimal three-guard boundary cover is simple (i.e., guard paths do not cross one another). We provide complete characterizations of the form of optimal two- and three-guard boundary covers for convex polygons that support polynomial-time algorithms for their construction. Finally, we show that, for convex P , any optimal two- or three-guard cover of @P is also a (necessarily optimal) cover of the full polygon P .
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".