Near-gathering of energy-constrained mobile agents

Abstract

We study the task of gathering k energy-constrained mobile agents in an undirected edgeweighted graph. Each agent is initially placed on an arbitrary node and has a limited amount of energy, which constrains the distance it can move. Since this may render gathering at a single point impossible, we study three variants of near-gathering: The goal is to move the agents into a configuration that minimizes either (i) the radius of a ball containing all agents, (ii) the maximum distance between any two agents, or (iii) the average distance between the agents. We prove that (i) is polynomial-time solvable, (ii) has a polynomial-time 2-approximation with a matching NP-hardness lower bound, while (iii) admits a polynomial-time 2(1 − 1 k)-approximation, but no FPTAS, unless P = NP. We extend some of our results to additive approximation.