Competitive Perimeter Defense in Tree Environments

TL;DR

This paper studies a perimeter defense problem in tree environments, analyzing online algorithms' performance against an optimal offline strategy, and proposes algorithms with provable competitiveness.

Contribution

It introduces fundamental limits on online algorithms' performance and designs three classes of algorithms with proven competitiveness in tree environments.

Findings

Derived conditions for finite, 2, and 3/2 competitive ratios.

Designed three online algorithms with provable competitiveness.

Numerical visualization of environmental regimes and algorithm performance.

Abstract

We consider a perimeter defense problem in a rooted full tree graph environment in which a single defending vehicle seeks to defend a set of specified vertices, termed as the perimeter from mobile intruders that enter the environment through the tree's leaves. We adopt the technique of competitive analysis to characterize the performance of an online algorithm for the defending vehicle. We first derive fundamental limits on the performance of any online algorithm relative to that of an optimal offline algorithm. Specifically, we give three fundamental conditions for finite, 2, and 3/2 competitive ratios in terms of the environment parameters. We then design and analyze three classes of online algorithms that have provably finite competitiveness under varying environmental parameter regimes. Finally, we give a numerical visualization of these regimes to better show the comparative strengths and weaknesses of each algorithm.