Complete Graph Identification in Population Protocols
Haruki Kanaya, Yuichi Sudo
TL;DR
The paper tackles complete-graph identification in population protocols under two fairness models. It introduces a global-fairness protocol $ exttt{CIG}$ with $O(n^2)$ states that identifies complete graphs without prior size knowledge, and two weak-fairness protocols, $ exttt{CIW}_n$ (with exact $n$) and its generalized $ exttt{CIW}_{n,k}$, achieving $O(n)$ and $O(nk2^k)$ states respectively while reporting the result efficiently. The work also proves an impossibility result: with only a common bound $P$ on $n$, complete-graph identification is not solvable under weak fairness. Collectively, these results characterize the trade-offs between prior knowledge, fairness, state complexity, and time, advancing understanding of graph-identification capabilities in population protocols.
Abstract
We consider the population protocol model where indistinguishable state machines, referred to as agents, communicate in pairs. The communication graph specifies potential interactions (\ie communication) between agent pairs. This paper addresses the complete graph identification problem, requiring agents to determine if their communication graph is a clique or not. We evaluate various settings based on: (i) the fairness preserved by the adversarial scheduler -- either global fairness or weak fairness, and (ii) the knowledge provided to agents beforehand -- either the exact population size $n$, a common upper bound $P$ on $n$, or no prior information. Positively, we show that $O(n^2)$ states per agent suffice to solve the complete graph identification problem under global fairness without prior knowledge. With prior knowledge of $n$, agents can solve the problem using only $O(n)$ states under weak fairness. Negatively, we prove that complete graph identification remains unsolvable under weak fairness when only a common upper bound $P$ on the population size $n$ is known.