Disk and Partial Disk Inspection: Worst- to Average-Case and Pareto Upper Bounds

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.