Disk and Partial Disk Inspection: Worst- to Average-Case and Pareto Upper Bounds
Bibliographic record
Abstract
We consider $n$ unit-speed mobile agents initially positioned at the center of a unit disk, tasked with inspecting all points on the disk's perimeter. A perimeter point is considered covered if an agent located outside the disk's interior has unobstructed visibility of it, treating the disk itself as an obstacle. For $n=1$, this problem is known as the shoreline problem with a known distance. Isbell (1957) derived an optimal trajectory that minimizes the worst-case inspection time for this problem, while Gluss (1961) proposed heuristics for its average-case version. The one-agent case was originally introduced as a more tractable variant of Bellman's famous lost-in-the-forest problem. Our contributions are threefold. First, as a warm-up, we extend Isbell's findings by deriving worst-case optimal trajectories for partial inspection of a section of the disk, thereby providing an alternative proof of optimality for inspection with $n \geq 2$ agents. Second, we improve Gluss's bounds on the average-case inspection time under a uniform distribution of perimeter points (equivalent to randomized inspection algorithms), and we also strengthen the methodology by combining spatial discretization with Nonlinear Programming (NLP) to build feasible solutions to the continuous problem and compare them with NLP solutions. Third, we establish Pareto-optimal bounds for the multi-objective problem of jointly minimizing the worst-case and average-case inspection times.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".