On Optimal Coverage of a Tree with Multiple Robots

TL;DR

This paper investigates the problem of optimally covering a tree with multiple robots, analyzing different objectives and constraints, and establishing the computational complexity of these problems.

Contribution

It provides complexity results for covering a tree with multiple robots under various objectives and rendezvous constraints, including NP-hardness and polynomial-time solvability.

Findings

Cover time minimization is NP-hard when the number of robots is part of the input.

Cover length minimization is polynomial-time solvable without rendezvous constraints.

Cover length minimization becomes NP-hard when periodic rendezvous are required.

Abstract

We study the algorithmic problem of optimally covering a tree with $k$ mobile robots. The tree is known to all robots, and our goal is to assign a walk to each robot in such a way that the union of these walks covers the whole tree. We assume that the edges have the same length, and that traveling along an edge takes a unit of time. Two objective functions are considered: the cover time and the cover length. The cover time is the maximum time a robot needs to finish its assigned walk and the cover length is the sum of the lengths of all the walks. We also consider a variant in which the robots must rendezvous periodically at the same vertex in at most a certain number of moves. We show that the problem is different for the two cost functions. For the cover time minimization problem, we prove that the problem is NP-hard when $k$ is part of the input, regardless of whether periodic rendezvous are required or not. For the cover length minimization problem, we show that it can be solved in polynomial time when periodic rendezvous are not required, and it is NP-hard otherwise.