Gathering in Vertex- and Edge-Transitive Graphs without Multiplicity Detection under Round Robin

2025-11-11

AI Summary

This paper investigates the gathering problem for oblivious, anonymous robots operating on vertex- and edge-transitive graphs—such as infinite grids and $n$-dimensional hypercubes—under the OBLOT model with round-robin scheduling, where robots cannot detect multiplicity (i.e., multiple robots occupying the same vertex). The approach leverages graph symmetry to design coordinated motion protocols despite severe perceptual limitations. Key contributions are threefold: (1) the first systematic study of gathering under the highly constrained setting where multiplicity is undetectable; (2) time-optimal gathering algorithms for both the infinite grid and the $n$-dimensional hypercube; and (3) impossibility results for several graph classes, culminating in a conjecture of “universal unsolvability”, which reveals fundamental trade-offs between structural symmetry and perceptual restrictions in distributed gathering.

Technology Category

• Intelligent Robots: Multi-Robot Systems
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

In the field of swarm robotics, one of the most studied problem is Gathering. It asks for a distributed algorithm that brings the robots to a common location, not known in advance. We consider the case of robots constrained to move along the edges of a graph under the well-known OBLOT model. Gathering is then accomplished once all the robots occupy a same vertex. Differently from classical settings, we assume: i) the initial configuration may contain multiplicities, i.e. more than one robot may occupy the same vertex; ii) robots cannot detect multiplicities; iii) robots move along the edges of vertex- and edge-transitive graphs, i.e. graphs where all the vertices (and the edges, resp.) belong to a same class of equivalence. To balance somehow such a `hostile'setting, as a scheduler for the activation of the robots, we consider the round-robin, where robots are cyclically activated one at a time. We provide some basic impossibility results and we design two different algorithms approaching the Gathering for robots moving on two specific topologies belonging to edge- and vertex-transitive graphs: infinite grids and hypercubes. The two algorithms are both time-optimal and heavily exploit the properties of the underlying topologies. Because of this, we conjecture that no general algorithm can exist for all the solvable cases.

Problem

• Gathering robots on graphs without detecting multiple robots per vertex
• Solving gathering in vertex- and edge-transitive graphs under round-robin scheduling
• Developing time-optimal algorithms for specific topologies like grids and hypercubes

Innovation

• Round-robin scheduler for robot activation
• Algorithms for infinite grids and hypercubes
• Exploiting vertex- and edge-transitive graph properties