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：我们研究一个组合优化问题，涉及使用单位速度的智能体巡逻锐角三角形的边。目标是最大化（1-间隙）任意边的最大空闲时间，即该边连续两次被访问之间的时间间隔。该问题源于1775年法尼亚诺提出的一个历时数百年的优化问题，他试图找到具有最小周长的锐角三角形内接三角形。众所周知，垂足三角形（其产生遵循几何光学定律的周期循环轨迹）是法尼亚诺问题的最优解。此类轨迹被称为法尼亚诺轨道，或更一般地称为台球轨迹。我们证明垂足三角形也是该巡逻问题的最优解。我们的主要贡献在于揭示了台球轨迹与组合优化中最佳巡逻调度之间的新联系。特别地，作为我们论证的副产品，我们引入了一个新颖的2-间隙巡逻问题，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结构良好的台球型最优轨迹，包括垂足轨迹——其具有最小化任意两条连续访问边之间访问时间间隔的特殊性质。与之互补的是，我们还研究了动态次优轨迹在1-间隙巡逻优化问题中的代价。这些轨迹由贪心算法产生，可通过计算能力原始的移动智能体实现。