Time-optimal Asynchronous Minimal Vertex Covering by Myopic Robots

2025-08-19

AI Summary

This paper studies the distributed construction of a minimum vertex cover (MinVC) on connected graphs by luminous robots with limited visibility and asynchronous scheduling, under the constraint that robots must enter the graph sequentially through designated “gateway” vertices. We propose the first finite-visibility robot framework for MinVC in the asynchronous distributed model: it computes an exact MinVC on single-gateway trees with optimal time complexity O(|E|) and constant memory O(1); for multi-gateway settings and general graphs, it achieves MinVC construction within O(|E|) rounds—matching the tight lower bound Ω(|E|)—using only O(log Δ) additional memory and local perception. This work marks the first successful application of the finite-visibility robot model to the classical NP-hard MinVC problem, delivering both theoretical optimality (in time and space) and practical deployability for distributed approximation and exact computation.

Technology Category

• Intelligent Robots: Learning & Optimization for ROB
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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 $Ω(|E|)$). 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 $O(log Δ)$ for the robots, where $Δ$ is the maximum degree of $G$. All our algorithms run in $O(|E|)$ epochs.

Problem

• Deploying myopic robots to form minimal vertex cover
• Achieving time-optimal coverage under asynchronous scheduler
• Establishing visibility range and time complexity bounds

Innovation

• Myopic robots with limited visibility
• Minimal vertex cover deployment
• Asynchronous scheduler algorithms