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Optimal Average Disk-Inspection via Fermat’s Principle

Author Konstantinos Georgiou

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LIPIcs.STACS.2026.44.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Theory and algorithms for application domains
Keywords
  • Inspection
  • Disk
  • Average-Case Performance

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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 [Isbell, 1957], but the average-case version remained open, with heuristic upper bounds proposed by Gluss [Gluss, 1961] in 1961 and improved only recently in [Conley and Georgiou, 2025]. 
Our approach applies Fermat’s Principle of Least Time from optics to the discretization framework of [Conley and Georgiou, 2025], 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 [Gluss, 1961] that optimal trajectories must touch the disk. 
Our analysis determines the exact optimal average-case inspection cost, equal to 3.549259… and certified to at least six digits of accuracy.

Cite As Get BibTex

Konstantinos Georgiou. Optimal Average Disk-Inspection via Fermat’s Principle. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.44

Author Details

Konstantinos Georgiou
  • Toronto Metropolitan University, Canada

Funding

Research supported in part by an NSERC Discovery Grant and by the Toronto Metropolitan University Faculty of Science Dean’s Research Fund.

Acknowledgements

We thank Derek Muller and his team at Veritasium for an expository discussion of Fermat’s principle of least time, in particular the episode "The Closest We’ve Come to a Theory of Everything", without which the perspective adopted in this work would likely not have emerged. Special thanks also to Caleb Jones for early discussions of the project.

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