Population Protocols for Graph Class Identification Problems

TL;DR

This paper investigates the capabilities and limitations of population protocols in identifying various graph classes, proposing protocols for some classes under certain conditions and proving impossibility results for others.

Contribution

It introduces new population protocols for identifying specific graph classes and establishes impossibility results under different assumptions.

Findings

Protocols for $k$-regular graphs and trees under global fairness.

Protocol for stars under weak fairness.

Impossibility of protocols for several classes under weak or arbitrary initial states.

Abstract

In this paper, we focus on graph class identification problems in the population protocol model. A graph class identification problem aims to decide whether a given communication graph is in the desired class (e.g. whether the given communication graph is a ring graph). Angluin et al. proposed graph class identification protocols with directed graphs and designated initial states under global fairness [Angluin et al., DCOSS2005]. We consider graph class identification problems for undirected graphs on various assumptions such as initial states of agents, fairness of the execution, and initial knowledge of agents. In particular, we focus on lines, rings, $k$-regular graphs, stars, trees, and bipartite graphs. With designated initial states, we propose graph class identification protocols for $k$-regular graphs, and trees under global fairness, and propose a graph class identification protocol for stars under weak fairness. Moreover, we show that, even if agents know the number of agents $n$, there is no graph class identification protocol for lines, rings, $k$-regular graphs, trees, or bipartite graphs under weak fairness. On the other hand, with arbitrary initial states, we show that there is no graph class identification protocol for lines, rings, $k$-regular graphs, stars, trees, or bipartite graphs.