Optimal Average Disk-Inspection via Fermat's Principle

2025-09-08

AI Summary

This paper resolves the long-standing open problem of determining the optimal average detection cost for the Disk-Inspection problem originally posed by Bellman (1955). The problem asks for the minimal average time required by a mobile agent, starting at the center of a unit disk, to detect an unknown boundary point. While the worst-case variant was solved by Isbell (1957), the average-case version—open since Gluss’s (1961) heuristic upper bound—remained unsolved. Methodologically, we integrate Fermat’s principle of least time with discrete approximation to formulate a single-parameter optimal control model in the continuous limit; leveraging an optical analogy, we rigorously reduce dimensionality and prove that the optimal trajectory never contacts the disk boundary—refuting Gluss’s conjecture. We fully characterize the trajectory’s geometric structure and compute the exact optimal average cost as 3.549259… (to six decimal places), thereby settling this sixty-year-old problem.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Web measurements

Abstract

This work resolves the optimal average-case cost of the Disk-Inspection problem, a variant of Bellman's 1955 lost-in-a-forest problem. In Disk-Inspection, a mobile agent starts at the center of a unit disk and follows a trajectory that inspects perimeter points whenever the disk does not obstruct visibility. The worst-case cost was solved optimally in 1957 by Isbell, but the average-case version remained open, with heuristic upper bounds proposed by Gluss in 1961 and improved only recently. Our approach applies Fermat's Principle of Least Time to a recently proposed discretization framework, showing that optimal solutions are captured by a one-parameter family of recurrences independent of the discretization size. In the continuum limit these recurrences give rise to a single-parameter optimal control problem, whose trajectories coincide with limiting solutions of the original Disk-Inspection problem. A crucial step is proving that the optimal initial condition generates a trajectory that avoids the unit disk, thereby validating the optics formulation and reducing the many-variable optimization to a rigorous one-parameter problem. In particular, this disproves Gluss's conjecture that optimal trajectories must touch the disk. Our analysis determines the exact optimal average-case inspection cost, equal to $3.549259ldots$ and certified to at least six digits of accuracy.

Problem

• Determining optimal average-case cost for Disk-Inspection problem
• Resolving open question about perimeter inspection trajectories
• Disproving Gluss's conjecture about optimal disk contact

Innovation

• Applied Fermat's Principle to disk-inspection discretization
• Reduced optimization to single-parameter control problem
• Proved optimal trajectories avoid disk, disproving prior conjecture