A Dynamic Boundary Guarding Problem with Translating Targets

TL;DR

This paper studies a dynamic boundary guarding problem where a service vehicle aims to intercept mobile targets approaching a deadline boundary, proposing optimal and near-optimal policies for different speed regimes and analyzing their performance.

Contribution

It introduces novel vehicle policies for guarding a boundary against moving targets, including optimal solutions for certain regimes and performance bounds.

Findings

The longest path policy is optimal when the distance to the deadline is large.

The proposed policies achieve near-optimal capture fractions in their respective regimes.

Performance bounds are established for different target and vehicle speed scenarios.

Abstract

We introduce a problem in which a service vehicle seeks to guard a deadline (boundary) from dynamically arriving mobile targets. The environment is a rectangle and the deadline is one of its edges. Targets arrive continuously over time on the edge opposite the deadline, and move towards the deadline at a fixed speed. The goal for the vehicle is to maximize the fraction of targets that are captured before reaching the deadline. We consider two cases; when the service vehicle is faster than the targets, and; when the service vehicle is slower than the targets. In the first case we develop a novel vehicle policy based on computing longest paths in a directed acyclic graph. We give a lower bound on the capture fraction of the policy and show that the policy is optimal when the distance between the target arrival edge and deadline becomes very large. We present numerical results which suggest near optimal performance away from this limiting regime. In the second case, when the targets are slower than the vehicle, we propose a policy based on servicing fractions of the translational minimum Hamiltonian path. In the limit of low target speed and high arrival rate, the capture fraction of this policy is within a small constant factor of the optimal.