Run for Cover: Dominating Set via Mobile Agents

TL;DR

This paper introduces algorithms for mobile agents to find minimal and near-minimal dominating sets in graphs, enabling efficient coverage and guarding strategies in robotic and network systems.

Contribution

It presents the first algorithms for minimal dominating sets using mobile agents, with complexity bounds for both single and multiple starting positions.

Findings

Algorithms for minimal dominating set with linear and near-linear time complexity.

A logarithmic approximation algorithm for the minimum dominating set.

Applicability to robotic coverage and network guarding problems.

Abstract

Research involving computing with mobile agents is a fast-growing field, given the advancement of technology in automated systems, e.g., robots, drones, self-driving cars, etc. Therefore, it is pressing to focus on solving classical network problems using mobile agents. In this paper, we study one such problem -- finding small dominating sets of a graph $G$ using mobile agents. Dominating set is interesting in the field of mobile agents as it opens up a way for solving various robotic problems, e.g., guarding, covering, facility location, transport routing, etc. In this paper, we first present two algorithms for computing a {\em minimal dominating set}: (i) an $O(m)$ time algorithm if the robots start from a single node (i.e., gathered initially), (ii) an $O(\ell\Delta\log(\lambda)+n\ell+m)$ time algorithm, if the robots start from multiple nodes (i.e., positioned arbitrarily), where $m$ is the number of edges and $\Delta$ is the maximum degree of $G$, $\ell$ is the number of clusters of the robot initially and $\lambda$ is the maximum ID-length of the robots. Then we present a $\ln (\Delta)$ approximation algorithm for the {\em minimum} dominating set which takes $O(n\Delta\log (\lambda))$ rounds.