Optimal patrolling of fragmented boundaries

Abstract

A set of mobile robots is deployed on a simple curve of finite length, composed of a finite set of vital segments separated by neutral segments. The robots have to patrol the vital segments by perpetually moving on the curve, without exceeding their maximum speed. The quality of patrolling is measured by the idleness, i.e., the longest time period during which any vital point on the curve is not visited by any robot. Given a configuration of vital segments, our goal is to provide algorithms describing the movement of the robots along the curve so as to minimize the idleness.Our main contribution is a proof that the optimal solution to the patrolling problem is attained either by the cyclic strategy, in which all the robots move in one direction around the curve, or by the partition strategy, in which the curve is partitioned into sections which are patrolled separately by individual robots. These two fundamental types of strategies were studied in the past in the robotics community in different theoretical and experimental settings. However, to our knowledge, this is the first theoretical analysis proving optimality in such a general scenario. Throughout the paper we assume that all robots have the same maximum speed. In fact, the claim is known to be invalid when this assumption does not hold, cf. [Czyzowicz et al., Proc. ESA 2011].

Patrolling a Border

“…In the patrolling literature, for example in Collins et al (2013) an important qualitative question is whether the network under attack can be defended with non overlapping patrols, either by separate patrollers (when there are several) or when the patrols are mixed probabilistically.…”

Optimizing periodic patrols against short attacks on the line and other networks

“…Multiple infiltrators are also considered by Zoroa et al (2012) where the infiltration is through a circular rather than a linear boundary. Multiple patrollers, when only some portions of the boundary need to be protected, are considered by Collins et al (2013), who show how the problem can be divided up. Papadaki et al (2016) consider the discrete border patrol problem, where the infiltration can only be accomplished at certain points of the border (perhaps mountain passes).…”

Computational geometry column 59

“…Given n stations on the unit circle, each with a prescribed maximum idle time, and k mobile agents, each with a given maximum speed, is there a schedule for the agents such that each station remains unattended for at most its prescribed maximum idle time? Observe that the problem is non-trivial only if k < n. More generally, the n stations can be intervals on the unit circle, as in [6].…”

“…Multi-agent patrolling is a variation of the problem of multi-robot coverage [4,5], studied extensively in the robotics community. A variety of models has been considered for patrolling, including deterministic and randomized, as well as centralized and distributed, under various objectives [1,6,11]. Idleness, as a measure of efficiency for a patrolling strategy, was introduced by Machado et al [13] in a graph setting; see also the article by Chevaleyre [4].…”