On cooperative patrolling: optimal trajectories, complexity analysis, and approximation algorithms

TL;DR

This paper investigates optimal patrolling strategies using autonomous agents across different graph structures, providing algorithms for chains and trees, and an approximation for cyclic graphs, with analysis of complexity and distributed control.

Contribution

It introduces polynomial algorithms for chain and tree graphs, and an NP-hardness proof with an approximation algorithm for cyclic graphs in patrolling tasks.

Findings

Polynomial time algorithms for chain and tree graphs.

NP-hardness of cyclic graph patrolling problem.

Constant factor approximation algorithm for cyclic graphs.

Abstract

The subject of this work is the patrolling of an environment with the aid of a team of autonomous agents. We consider both the design of open-loop trajectories with optimal properties, and of distributed control laws converging to optimal trajectories. As performance criteria, the refresh time and the latency are considered, i.e., respectively, time gap between any two visits of the same region, and the time necessary to inform every agent about an event occurred in the environment. We associate a graph with the environment, and we study separately the case of a chain, tree, and cyclic graph. For the case of chain graph, we first describe a minimum refresh time and latency team trajectory, and we propose a polynomial time algorithm for its computation. Then, we describe a distributed procedure that steers the robots toward an optimal trajectory. For the case of tree graph, a polynomial time algorithm is developed for the minimum refresh time problem, under the technical assumption of a constant number of robots involved in the patrolling task. Finally, we show that the design of a minimum refresh time trajectory for a cyclic graph is NP-hard, and we develop a constant factor approximation algorithm.