Randomized Competitive Perimeter Defense on a Line

We consider a perimeter defense problem in which a vehicle seeks to defend a compact region from mobile intruders in a one-dimensional environment parameterized by the perimeter size relative to the environment and the intruder-to-vehicle speed ratio. The intruders move with fixed speed and direction to reach the perimeter. We present a competitive analysis approach to this problem by measuring the performance of randomized 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 characterize regimes in the parameter space in which, for any online randomized algorithm, (i) the competitive ratio has to be at least 2 and (ii) the competitive ratio has to be at least 1.33. We then design three randomized algorithms and characterize their competitive ratios. Finally, we present parameter regime plots that provide insights into parameter ranges in which the algorithms’ performances are near optimal.