Competitive Perimeter Defense of Conical Environments

TL;DR

This paper analyzes online algorithms for perimeter defense in conical environments, establishing conditions for finite competitiveness and designing algorithms with provable competitive ratios.

Contribution

It introduces a competitive analysis framework for perimeter defense in conical environments and develops algorithms with guaranteed performance bounds.

Findings

Two necessary conditions for finite competitiveness.

Designed three online algorithms with competitive ratios of 1 and 2.

Numerical analysis reveals parameter regimes for algorithm performance.

Abstract

We consider a perimeter defense problem in a planar conical environment in which a single vehicle, having a finite capture radius, aims to defend a concentric perimeter from mobile intruders. The intruders are arbitrarily released at the circumference of the environment and they move radially toward the perimeter with fixed speed. We present a competitive analysis approach to this problem by measuring the performance of multiple online algorithms for the vehicle against arbitrary inputs, relative to an optimal offline algorithm that has information about entire input sequence in advance. In particular, we establish two necessary conditions on the parameter space to guarantee (i) finite competitiveness of any algorithm and (ii) a competitive ratio of at least 2 for any algorithm. We then design and analyze three online algorithms and characterize parameter regimes in which they have finite competitive ratios. Specifically, our first two algorithms are provably 1, and 2-competitive, respectively, whereas our third algorithm exhibits different competitive ratios in different regimes of problem parameters. Finally, we provide a numerical plot in the parameter space to reveal additional insights into the relative performance of our algorithms.