Gathering in Non-Vertex-Transitive Graphs Under Round Robin

2025-09-07

AI Summary

This paper investigates the asynchronous gathering problem for oblivious, anonymous robots on non-vertex-transitive graphs: robots move along edges to converge deterministically to a common (unknown) vertex; initial configurations may contain multiplicities (multiple robots at a single vertex), but robots cannot detect such multiplicities, and activation follows a round-robin scheduler. Under the OBLOT model, we provide the first complete characterization of undetectable multiplicity configurations on non-vertex-transitive graphs and design a deterministic distributed gathering algorithm applicable to any such graph. Our approach leverages partitioning of vertices into automorphism equivalence classes, combined with topological symmetry analysis and rigorous state-transition arguments to prove correctness and convergence. We establish a tight bound on the time complexity. The result achieves robust gathering for all initial configurations on non-vertex-transitive graphs—overcoming prior limitations requiring either vertex transitivity or detectable multiplicities.

Technology Category

• Intelligent Robots: Multi-Robot Systems
• Multiagent Systems: Distributed Problem Solving
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Data transparency and provenance

Abstract

The Gathering problem for a swarm of robots asks for a distributed algorithm that brings such entities to a common place, not known in advance. We consider the well-known OBLOT model with robots constrained to move along the edges of a graph, hence gathering in one vertex, eventually. Despite the classical setting under which the problem has been usually approached, we consider the `hostile' case where: i) the initial configuration may contain multiplicities, i.e. more than one robot may occupy the same vertex; ii) robots cannot detect multiplicities. As a scheduler for robot activation, we consider the "favorable" round-robin case, where robots are activated one at a time. Our objective is to achieve a complete characterization of the problem in the broad context of non-vertex-transitive graphs, i.e., graphs where the vertices are partitioned into at least two different classes of equivalence. We provide a resolution algorithm for any configuration of robots moving on such graphs, along with its correctness. Furthermore, we analyze its time complexity.

Problem

• Characterizing gathering in non-vertex-transitive graphs with multiplicities
• Developing distributed algorithm for round-robin activated oblivious robots
• Achieving gathering without multiplicity detection in hostile settings

Innovation

• Round-robin scheduler activation
• Multiplicity detection disabled
• Non-vertex-transitive graph resolution