The dispersion involves the coordination of $k \leq n$ agents on a graph of size $n$ to reach a configuration where at each node at most one agent can be present. It is a well-studied problem. Also, this problem is studied on dynamic graphs with $n$ nodes where at each discrete time step the graph is a connected sub-graph of the complete graph $K_n$. An optimal algorithm is provided assuming global communication and 1-hop visibility of the agents. How this problem pans out on Time-Varying Graphs (TVG) is an open question in the literature. In this work we study this problem on TVG where at each discrete time step the graph is a connected sub-graph of an underlying graph $G$ (known as a footprint) consisting of $n$ nodes. We have the following results even if only one edge from $G$ is missing in the connected sub-graph at any time step and all agents start from a rooted initial configuration. Even with unlimited memory at each agent and 1-hop visibility, it is impossible to solve dispersion for $n$ co-located agents on a TVG in the local communication model. Furthermore, even with unlimited memory at each agent but without 1-hop visibility, it is impossible to achieve dispersion for $n$ co-located agents in the global communication model. From the positive side, the existing algorithm for dispersion on dynamic graphs with the assumptions of global communication and 1-hop visibility works on TVGs as well. This fact and the impossibility results push us to come up with a modified definition of the dispersion problem on TVGs, as one needs to start with more than $n$ agents if the objective is to drop the strong assumptions of global communication and 1-hop visibility. Then, we provide an algorithm to solve the modified dispersion problem on TVG starting with $n+1$ agents with $O(\log n)$ memory per agent while dropping both the assumptions of global communication and 1-hop visibility.

翻译：分散问题涉及在规模为$n$的图上协调$k \leq n$个智能体，以达到每个节点至多存在一个智能体的配置状态。这是一个被深入研究的问题。该问题同样在动态图上被探讨：在包含$n$个节点的动态图中，每个离散时间步的图是完全图$K_n$的连通子图。在假设全局通信和智能体具备单跳可见性的前提下，已有最优算法被提出。该问题在时变图上的表现是文献中一个悬而未决的问题。本文研究时变图上的分散问题，其中每个离散时间步的图是由$n$个节点组成的底层图$G$（称为足迹图）的一个连通子图。即使在任何时间步，连通子图中仅缺失$G$的一条边，且所有智能体均从根初始配置出发，我们仍得到以下结果：在局部通信模型中，即使每个智能体拥有无限内存和单跳可见性，也无法在时变图上解决$n$个共置智能体的分散问题。此外，在全局通信模型中，即使每个智能体拥有无限内存但无单跳可见性，同样无法实现$n$个共置智能体的分散。从积极方面看，现有基于全局通信和单跳可见性假设的动态图分散算法在时变图上同样有效。这一事实与不可能性结果促使我们对时变图上的分散问题进行修正定义：若目标是摒弃全局通信和单跳可见性的强假设，则需要以多于$n$个智能体作为初始条件。随后，我们提出一种算法，以$n+1$个智能体为起点，在时变图上解决修正后的分散问题，该算法仅需每个智能体$O(\log n)$的内存，同时摒弃了全局通信和单跳可见性两项假设。