Semi-Synchronous Exploration in Dynamic Graphs

TL;DR

This paper investigates the limits of graph exploration by mobile agents in dynamic, semi-synchronous networks under adversarial deactivation, establishing bounds and proposing an optimal exploration algorithm.

Contribution

It determines the maximum number of agents that can be deactivated per round for exploration to be possible and provides an algorithm that achieves exploration at this threshold.

Findings

Exploration is impossible if the adversary deactivates at least ⌈k/(n-2)⌉ - 1 agents per round.

Exploration is feasible only if the adversary deactivates at most ⌈k/(n-2)⌉ - 2 agents per round.

The proposed exploration algorithm matches the derived deactivation bound under certain visibility and communication assumptions.

Abstract

We study the fundamental problem of graph exploration in dynamic graphs using mobile agents. We consider $1$-interval connected dynamic graphs, where the topology may change arbitrarily from round to round as long as the graph remains connected, and edges are assigned with the dynamic port labeling at each round. The execution follows a semi-synchronous scheduler, under which an adversary may deactivate an arbitrary subset of agents in each round. For a graph with $n$ nodes and $k$ agents, we show that exploration is impossible if the adversary can deactivate at least $ \left\lceil \frac{k}{n-2} \right\rceil - 1$ agents per round, even when agents are equipped with unbounded memory, have global communication and full visibility. This yields an upper bound, implying that exploration is solvable only when the adversary deactivates at most $\left\lceil \frac{k}{n-2} \right\rceil - 2$ agents per round. We further establish that achieving exploration at this threshold requires agents to have both $1$-hop visibility and $1$-hop communication. Finally, we present the exploration algorithm using $k$ agents when the adversary deactivates at most $ \left\lceil \frac{k}{n-2} \right\rceil - 2$ agents, assuming agents are equipped with $1$-hop visibility and global communication, and matches the adversarial deactivation bound implied by the impossibility results.