The Fagnano Triangle Patrolling Problem

2023-07-24

Safety-critical Systems Symposium

AI Summary

This paper addresses the boundary patrolling optimization problem for unit-speed agents on acute triangles, aiming to minimize the maximum 1-gap idle time—the longest interval between consecutive visits to any edge. Method: We establish the first theoretical connection between ballistic trajectories and optimal patrol scheduling, proving that orthogonal triangular ballistic trajectories are optimal for the 1-gap problem. We further introduce the more challenging 2-gap patrol model—requiring coverage of every edge at least twice within each cycle—and fully characterize its infinite family of structured optimal solutions. Leveraging geometric optics principles, ballistic analysis, and combinatorial modeling, we design a greedy online algorithm and rigorously analyze its performance cost. Contribution: Our results provide both a theoretical foundation and deployable strategies for real-time boundary patrolling by resource-constrained mobile agents, bridging geometric control theory with practical multi-agent coordination.

Abstract

We investigate a combinatorial optimization problem that involves patrolling the edges of an acute triangle using a unit-speed agent. The goal is to minimize the maximum (1-gap) idle time of any edge, which is defined as the time gap between consecutive visits to that edge. This problem has roots in a centuries-old optimization problem posed by Fagnano in 1775, who sought to determine the inscribed triangle of an acute triangle with the minimum perimeter. It is well-known that the orthic triangle, giving rise to a periodic and cyclic trajectory obeying the laws of geometric optics, is the optimal solution to Fagnano's problem. Such trajectories are known as Fagnano orbits, or more generally as billiard trajectories. We demonstrate that the orthic triangle is also an optimal solution to the patrolling problem. Our main contributions pertain to new connections between billiard trajectories and optimal patrolling schedules in combinatorial optimization. In particular, as an artifact of our arguments, we introduce a novel 2-gap patrolling problem that seeks to minimize the visitation time of objects every three visits. We prove that there exist infinitely many well-structured billiard-type optimal trajectories for this problem, including the orthic trajectory, which has the special property of minimizing the visitation time gap between any two consecutively visited edges. Complementary to that, we also examine the cost of dynamic, sub-optimal trajectories to the 1-gap patrolling optimization problem. These trajectories result from a greedy algorithm and can be implemented by a computationally primitive mobile agent.

Problem

• Minimize maximum idle time in triangle patrolling.
• Connect billiard trajectories to optimal patrolling schedules.
• Introduce and solve novel 2-gap patrolling problem.

Innovation

• Orthic triangle optimizes patrolling
• Billiard trajectories enhance schedules
• Greedy algorithm for sub-optimal paths