|Résumé||A set of k mobile agents are placed on the boundary of a simply connected planar object represented by a cycle of unit length. Each agent has its own predefined maximal speed, and is capable of moving around this boundary with a velocity not exceeding its maximal speed. The agents are required to protect the boundary from an intruder which attempts to penetrate to the interior of the object through a point of the boundary, unknown to the agents. The intruder needs some time interval of length au to accomplish the intrusion. Will the intruder be able to penetrate into the object, or is there an algorithm allowing the agents to move perpetually along the boundary, so that no point of the boundary remains unprotected for a time period of au?
Such a problem may be solved by designing an algorithm which defines the motion of agents so as to minimize the idle time I, i.e., the longest time interval during which any fixed boundary point remains unvisited by some agent, with the obvious goal of achieving I < au.
Depending on the type of the environment, this problem is known as either "boundary patrolling" or "fence patrolling" in the robotics literature. The most common heuristics adopted in the past include the cyclic strategy, where agents move in one direction around the cycle covering the environment, and the partition strategy, in which the environment is partitioned
into sections patrolled separately by individual agents. This paper is, to our knowledge, the first study of the fundamental problem of boundary patrolling by agents with distinct maximal speeds. In this scenario, we give special attention to the performance of the cyclic strategy and the partition strategy. We propose general bounds and methods for analyzing these strategies, obtaining exact results for cases with 2, 3, and 4 agents. We show that there are cases when the cyclic strategy is optimal, cases when the partition strategy is optimal and, perhaps more surprisingly, cases when novel, alternative methods have to be used to achieve optimality.
This is joint work with Jurek Czyzowicz, Leszek Gasieniec and Evangelos Kranakis. |