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.

翻译：本文研究一个组合优化问题，涉及使用单位速度智能体对锐角三角形的边进行巡逻。目标是最大化减少任意边的最大（1间隙）空闲时间，该时间定义为连续访问该边之间的时间间隔。该问题源于1775年法尼亚诺提出的一个具有数百年历史的优化问题，即寻求锐角三角形中周长最小的内接三角形。众所周知，满足几何光学定律的周期性循环轨迹——垂足三角形，是法尼亚诺问题的最优解。此类轨迹被称为法尼亚诺轨道，或更一般地，台球轨迹。我们证明垂足三角形也是巡逻问题的一个最优解。主要贡献在于揭示了台球轨迹与组合优化中最优巡逻调度之间的新联系。特别地，作为论证的副产品，我们提出了一种新颖的2间隙巡逻问题，该问题旨在最小化每三次访问中物体的访问时间。我们证明该问题存在无穷多个具有良好结构的台球型最优轨迹，包括垂足轨迹——其特殊性质是能将任意两个连续访问边之间的访问时间差最小化。作为补充，我们还分析了动态次优轨迹在1间隙巡逻优化问题中的代价。这些轨迹源自贪心算法，可由计算能力原始的移动智能体实现。