Multivehicle Perimeter Defense in Conical Environments

In this article, we consider a perimeter defense problem in a planar conical environment in which
$M$
identical vehicles, each having a finite capture radius, seek to defend a concentric perimeter from mobile intruders. The intruders are released at the circumference of the environment at arbitrary times and in any number. Upon release, each intruder moves radially toward the perimeter with fixed speed. We provide a worst-case analysis of this problem. Specifically, we present a
competitive analysis
approach to this problem by measuring the performance of decentralized and cooperative online algorithms for the vehicles against arbitrary inputs, relative to an optimal offline algorithm that has information about entire intruder release sequence in advance. We first establish a necessary condition on the problem parameters that guarantees finite competitiveness of
any
algorithm. We then design and analyze three decentralized and two cooperative online algorithms and characterize parameter regimes in which they have finite competitive ratios. Specifically, our first two decentralized algorithms are provably 1 and 2-competitive, respectively, whereas our third decentralized algorithm exhibits different competitive ratios in different regimes of problem parameters. Our first cooperative algorithm is 1.5-competitive and our second cooperative algorithm exhibits different competitive ratios in different regimes of problem parameters. Finally, we provide multiple numerical plots in the parameter space to reveal additional insights into the relative performance of our algorithms and discuss an extension to the case of heterogeneous vehicles.