Negative Instance for the Edge Patrolling Beacon Problem

TL;DR

This paper disproves a previous conjecture by constructing specific polygons where an infinite-strength magnetic beacon cannot capture a moving iron ball, challenging assumptions in the edge patrolling beacon problem.

Contribution

It provides the first counterexamples showing that a ball-capturing beacon strategy does not always exist, refuting prior conjectures.

Findings

Counterexamples in orthogonal polygons

Counterexamples in general-position polygons

Disproof of the conjecture that such strategies always exist

Abstract

Can an infinite-strength magnetic beacon always ``catch'' an iron ball, when the beacon is a point required to be remain nonstrictly outside a polygon, and the ball is a point always moving instantaneously and maximally toward the beacon subject to staying nonstrictly within the same polygon? Kouhestani and Rappaport [JCDCG 2017] gave an algorithm for determining whether a ball-capturing beacon strategy exists, while conjecturing that such a strategy always exists. We disprove this conjecture by constructing orthogonal and general-position polygons in which the ball and the beacon can never be united.