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

INTEROPERABILITY FOR MODELING AND SIMULATION IN MARITIME EXTENDED FRAMEWORK

This thesis reports on the most relevant researches performed during the years of the Ph.D. at the Genova University and within the Simulation Team. The researches have been performed according to M&S well known recognized standards. The studies performed on interoperable simulation cover all the environments of the Extended Maritime Framework, namely Sea Surface, Underwater, Air, Coast & Land, Space and Cyber Space. The applications cover both the civil and defence domain. The aim is to demonstrate the potential of M&S applications for the Extended Maritime Framework, applied to innovative unmanned vehicles as well as to traditional assets, human personnel included. A variety of techniques and methodology have been fruitfully applied in the researches, ranging from interoperable simulation, discrete event simulation, stochastic simulation, artificial intelligence, decision support system and even human behaviour modelling

Optimization problems in infrastructure security

How do we identify and prioritize risks and make smart choices based on fiscal constraints and limited resources? The main goal of infrastructure security is to secure, withstand, and rapidly recover from potential threats that may affect critical resources located within a given bounded region. In order to strengthen and maintain secure, functioning, and resilient critical infrastructure, proactive and coordinated efforts are necessary. Motivated from questions raised by infrastructure security, in this paper we survey several recent optimization problems whose solution has occupied (and continues to occupy) computer science researchers in the last few years. Topics discussed include: 1. Patrolling. 2. Sensor Coverage and Interference. 3. Evacuation. 4. Domain Protection and Blocking. The central theme in all the problems mentioned above will involve mobility in that the participating agents will be able to move over a specified region with a given speed. Security in itself is undoubtedly a very broad and complex task which involves all layers of the communication process from physical to network. As such the limited goal of this survey is to outline existing models and ideas and discuss related open problems and future research directions, pertaining to optimization problems in infrastructure security