Competitive Perimeter Defense on a Line

TL;DR

This paper analyzes the effectiveness of different algorithms for a one-dimensional perimeter defense problem, establishing fundamental limits and designing algorithms with provable competitive ratios, supported by numerical studies.

Contribution

It introduces a competitive analysis framework for perimeter defense on a line, identifying limits and designing algorithms with specific competitive ratios for various parameter regimes.

Findings

No $c$-competitive algorithms exist for certain difficult parameters.

Designed algorithms with 1, 2, and 4-competitive ratios.

Numerical studies show algorithm performance against stochastic intruders.

Abstract

We consider a perimeter defense problem in which a single vehicle seeks to defend a compact region from intruders in a one-dimensional environment parameterized by the perimeter size and the intruder-to-vehicle speed ratio. The intruders move inward with fixed speed and direction to reach the perimeter. We provide both positive and negative worst-case performance results over the parameter space using competitive analysis. We first establish fundamental limits by identifying the most difficult parameter combinations that admit no $c$-competitive algorithms for any constant $c\geq 1$ and slightly easier parameter combinations in which every algorithm is at best $2$-competitive. We then design three classes of algorithms and prove they are $1$, $2$, and $4$-competitive, respectively, for increasingly difficult parameter combinations. Finally, we present numerical studies that provide insights into the performance of these algorithms against stochastically generated intruders.