Time-optimal Asynchronous Minimal Vertex Covering by Myopic Robots

TL;DR

This paper presents algorithms for myopic autonomous robots to efficiently find minimal and minimum vertex covers in graphs, including trees and general graphs, under asynchronous conditions with limited visibility.

Contribution

It introduces new algorithms for vertex cover deployment by myopic robots, achieving optimality in time and memory for trees and extending to general graphs with additional memory.

Findings

Algorithms run in O(|E|) epochs.

Single-door algorithms are time and memory optimal for trees.

Extended algorithms work for general graphs with O(log Δ) memory.

Abstract

In a connected graph with an autonomous robot swarm with limited visibility, it is natural to ask whether the robots can be deployed to certain vertices satisfying a given property using only local knowledge. This paper affirmatively answers the question with a set of \emph{myopic} (finite visibility range) luminous robots with the aim of \emph{filling a minimal vertex cover} (MVC) of a given graph $G = (V, E)$. The graph has special vertices, called \emph{doors}, through which robots enter sequentially. Starting from the doors, the goal of the robots is to settle on a set of vertices that forms a minimal vertex cover of $G$ under the asynchronous ($\mathcal{ASYNC}$) scheduler. We are also interested in achieving the \emph{minimum vertex cover} (MinVC, which is NP-hard \cite{Karp1972} for general graphs) for a specific graph class using the myopic robots. We establish lower bounds on the visibility range for the robots and on the time complexity (which is $\Omega(|E|)$). We present two algorithms for trees: one for single door, which is both time and memory-optimal, and the other for multiple doors, which is memory-optimal and achieves time-optimality when the number of doors is a constant. Interestingly, our technique achieves MinVC on trees with a single door. We then move to the general graph, where we present two algorithms, one for the single door and the other for the multiple doors with an extra memory of $O(\log \Delta)$ for the robots, where $\Delta$ is the maximum degree of $G$. All our algorithms run in $O(|E|)$ epochs.