Dynamic Boundary Guarding Against Radially Incoming Targets

We introduce a dynamic vehicle routing problem in which a single vehicle seeks to guard a circular perimeter against radially inward moving targets. Targets are generated uniformly as per a Poisson process in time with a fixed arrival rate on the boundary of a circle with a larger radius and concentric with the perimeter. Upon generation, each target moves radially inward toward the perimeter with a fixed speed. The aim of the vehicle is to maximize the capture fraction, i.e., the fraction of targets intercepted before they enter the perimeter. We first obtain a fundamental upper bound on the capture fraction which is independent of any policy followed by the vehicle. We analyze several policies in the low and high arrival rates of target generation. For low arrival, we propose and analyze a First-Come-First-Served and a Look-Ahead policy based on repeated computation of the path that passes through maximum number of unintercepted targets. For high arrival, we design and analyze a policy based on repeated computation of Euclidean Minimum Hamiltonian path through a fraction of existing targets and show that it is within a constant factor of the optimal. Finally, we provide a numerical study of the performance of the policies in parameter regimes beyond the scope of the analysis.