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

TL;DR

This work investigates the Gathering problem for oblivious robots moving on vertex- and edge-transitive graphs under Round Robin without multiplicity detection. It develops two time-optimal algorithms tailored to particular topologies: $\mathcal{A}_{\mathbf{H}}$ for hypercubes and $\mathcal{A}_{\mathbf{ST}}$ for infinite square grids, both leveraging topology-specific structures to break symmetries and drive convergence. The authors establish impossibility results for certain symmetric configurations and provide formal correctness proofs via task-based predicates and transition graphs, along with tight complexity bounds $O(d)$ for hypercubes and $O(n+m)$ for square grids. A key takeaway is the lack of a general universal strategy for all $\mathcal{VET}$ graphs under RR with no multiplicity detection, reinforcing the need for topology-aware approaches. The results advance understanding of gathering under highly symmetric conditions and may guide design of practical distributed robotic systems operating on regular networks.

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.

Figures (7)

• Figure 1: A representation configurations defined on hypercubes $Q_d$ with $d = 1$, $2$, $3$ and $4$. Black vertices represent robots. Dashed blue lines show the two possible partitions of $Q_2$ into hypercubes of dimension one.
• Figure 2: Ungatherable initial configurations for $Q_3$. Left: a $P_2$ graph, center: a $P_3$ graph, a fully occupied $Q_3$ graph.
• Figure 3: From left, configurations for which the function $\mathtt{DMA}$ returns false according to conditions (a), (b), (c), respectively.
• Figure 4: Example configurations. Left: a configuration in $T_{6}$, right: a configuration in $T_{7}$. The moving robot is highlighted with a circle. The arrow shows the movement direction.
• Figure 5: Configurations and moves planned by $\mathcal{A}_\mathbf{H}$ for Task $T_1$. The graph shows the transitions between configurations. Blue arrows show robot movements. Note that a self-loop exists in each node where a robot moves toward an occupied vertex, but they are omitted for presentation. Dashed arrows mean that the transitions are generated by robots moving from vertices occupied by multiplicities.

Theorems & Definitions (40)

• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• theorem 1.1
• proof
• theorem 1.2
• proof
• definition thmcounterdefinition
• theorem 1.3