Filling MIS Vertices by Myopic Luminous Robots

TL;DR

This paper introduces algorithms for luminous myopic robots to find a maximal independent set in a graph, with solutions tailored for single and multiple door entry scenarios under different scheduling and visibility conditions.

Contribution

It presents the first algorithms for MIS filling with luminous robots, addressing both single and multiple door cases with specific complexity bounds and robot capabilities.

Findings

Single door solution operates in O(n^2) epochs under asynchronous scheduling.

Multiple door solution operates in O(n^2) epochs under semi-synchronous scheduling.

Algorithms work with limited visibility, colors, and memory, advancing robot coordination in graph problems.

Abstract

We present the problem of finding a maximal independent set (MIS) (named as \emph{MIS Filling problem}) of an arbitrary connected graph having $n$ vertices with luminous myopic mobile robots. The robots enter the graph one after another from a particular vertex called the \emph{Door} and disperse along the edges of the graph without collision to occupy vertices such that the set of vertices occupied by the robots is a maximal independent set. We assume the robots have knowledge only about the maximum degree of the graph, denoted by $\Delta$. In this paper, we explore two versions of the problem: the solution to the first version, named as \emph{MIS Filling with Single Door}, works under an asynchronous scheduler using robots with 3 hops of visibility range, $\Delta + 6$ number of colors and $O(\log \Delta)$ bits of persistent storage. The time complexity is measured in terms of epochs and it can be solved in $O(n^2)$ epochs. An epoch is the smallest time interval in which each participating robot gets activated and executes the algorithm at least once. For the second version with $k~ ( > 1)$ \textit{Doors}, named as \emph{MIS Filling with Multiple Doors}, the solution works under a semi-synchronous scheduler using robots with 5 hops of visibility range, $\Delta + k + 6$ number of colors and $O(\log (\Delta + k))$ bits of persistent storage. The problem with multiple Doors can be solved in $O(n^2)$ epochs.