1,745 research outputs found
The Fagnano Triangle Patrolling Problem
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
Optimizing periodic patrols against short attacks on the line and other networks
On a given network, a Patroller and Attacker play the following win-lose game: The Patroller adopts a periodic walk on the network while the Attacker chooses a node and two consecutive periods (to attack there). The Patroller wins if he successfully intercepts the attack, that is, if he occupies the attacked node in one of the two periods of the attack. We solve this game in mixed strategies for line graphs, the first class of graphs to be solved for the periodic patrolling game. We also solve the game for arbitrary graphs when the period is even
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