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

Algorithmic and Combinatorial Results on Fence Patrolling, Polygon Cutting and Geometric Spanners

The purpose of this dissertation is to study problems that lie at the intersection of geometry and computer science. We have studied and obtained several results from three different areas, namely–geometric spanners, polygon cutting, and fence patrolling. Specifically, we have designed and analyzed algorithms along with various combinatorial results in these three areas. For geometric spanners, we have obtained combinatorial results regarding lower bounds on worst case dilation of plane spanners. We also have studied low degree plane lattice spanners, both square and hexagonal, of low dilation. Next, for polygon cutting, we have designed and analyzed algorithms for cutting out polygon collections drawn on a piece of planar material using the three geometric models of saw, namely, line, ray and segment cuts. For fence patrolling, we have designed several strategies for robots patrolling both open and closed fences

An extended study on addressing defender teamwork while accounting for uncertainty in attacker defender games using iterative Dec-MDPs

Multi-agent teamwork and defender-attacker security games are two areas that are currently receiving significant attention within multi-agent systems research. Unfortunately, despite the need for effective teamwork among multiple defenders, little has been done to harness the teamwork 1 research in security games. The problem that this paper seeks to solve is the coordination of decentralized defender agents in the presence of uncer-tainty while securing targets against an observing adversary. To address this problem, we offer the following novel contributions in this paper: (i) New model of security games with defender teams that coordinate under uncertainty; (ii) New algorithm based on column generation that uti-lizes Decentralized Markov Decision Processes (Dec-MDPs) to generate defender strategies that incorporate uncertainty; (iii) New techniques to handle global events (when one or more agents may leave the system) during defender execution; (iv) Heuristics that help scale up in the num-ber of targets and agents to handle real-world scenarios; (v) Exploration of the robustness of randomized pure strategies. The paper opens the door to a potentially new area combining computational game theory and multi-agent teamwork.

Improving Safety Service Patrol Performance

Safety Service Patrols (SSPs) provide motorists with assistance free of charge on most freeways and some key primary roads in Virginia. This research project is focused on developing a tool to help the Virginia Department of Transportation (VDOT) optimize SSP routes and schedules (hereafter called SSP-OPT). The computational tool, SSP-OPT, takes readily available data (e.g., corridor and segment lengths, turnaround points, average annual daily traffic) and outputs potential SSP configurations that meet the desired criteria and produce the best possible performance metrics for a given corridor. At a high level, the main components of the developed tool include capabilities to: a) generate alternative feasible SSP beat configurations for a corridor; b)predict incidents and SSP characteristics (e.g., incident frequency, SSP service time) for a given SSP beat configuration; c) estimate performance measures (e.g., SSP response time, number of incidents responded to); and d) identify and present the best SSP configuration(s) through visual aids that facilitate decision making. To generate the incident data needed for the simulation-based SSP-OPT tool, a hierarchical negative binomial model and a hierarchical Weibull model are developed for incident frequencies and incident durations, respectively, based on the historical incident data. These models have been found to be effective in simulating the spatiotemporal distribution of incidents along highway corridors and for generating their attribute data (e.g., incident type, duration). The simulation program employs a discrete event-based approach and requires a few calibration parameters (e.g., SSP vehicle speed). After calibrating the model, the validation results show good agreement with field observations when applied to a sample SSP corridor from I-95. A user interface is created for the SSP-OPT tool in MS Excel to facilitate data entry and visualization of the output metrics for a given corridor. The output includes the list of alternative feasible beat configurations and aggregated performance measures from multiple runs for each individual beat, as well as for each alternative beat configuration spanning the entire corridor. The proposed SSP optimization model could be applied to corridors with or without existing SSP service. The tool will help identify the best beat configurations to minimize SSP response times and maximize SSP response rates for a given number of SSP vehicles on a corridor. Implementing these optimal solutions in the field will result in travel time savings and improve highway safety since the SSP resources will be more efficiently utilized, thus reducing the impacts of incidents on traffic flow