Multi-Agent Online Graph Exploration on Cycles and Tadpole Graphs

TL;DR

The paper addresses multi-agent online exploration of unknown weighted graphs, focusing on cycles and tadpole graphs under time and energy costs. It introduces algorithmic strategies (ALE and AMP) that align online decisions with offline optima, achieving tight or near-tight competitive ratios: $1$-competitive on cycles in the energy model with $k=2$, $1$-competitive on tadpoles in the energy model with $k=3$, and $1.5$-competitive on tadpoles in the time model with $k=4$, with additional bounds for other agent counts. The work further extends results to $n$-tadpole graphs, showing $1$ energy with $n+2$ agents and $1.5+n/2$ time with the same, and provides $2^{n+1}$-agent extensions achieving $1.5$ time-competitive. These findings advance understanding of distributed exploration efficiency under online information revelation and various cost metrics, with implications for coordinating autonomous robotic teams in restricted unicyclic graphs.

Abstract

We study the problem of multi-agent online graph exploration, in which a team of k agents has to explore a given graph, starting and ending on the same node. The graph is initially unknown. Whenever a node is visited by an agent, its neighborhood and adjacent edges are revealed. The agents share a global view of the explored parts of the graph. The cost of the exploration has to be minimized, where cost either describes the time needed for the entire exploration (time model), or the length of the longest path traversed by any agent (energy model). We investigate graph exploration on cycles and tadpole graphs for 2-4 agents, providing optimal results on the competitive ratio in the energy model (1-competitive with two agents on cycles and three agents on tadpole graphs), and for tadpole graphs in the time model (1.5-competitive with four agents). We also show competitive upper bounds of 2 for the exploration of tadpole graphs with three agents, and 2.5 for the exploration of tadpole graphs with two agents in the time model.

Figures (13)

• Figure 1: Tadpole Graph Example. The paths $p_\ell, p_s, p_i, p_t$ and the edges $e_{mid}$ and $e_{max}$ are marked.
• Figure 2: A graph in which ALE has a competitive ratio of $2$ for the time model. The dashed lines are paths, consisting entirely of edges with length $\varepsilon$.
• Figure 3: Since $(s,x_1)$ is the longest edge, one agent moves clockwise from $s$ to $x_1$.
• Figure 4: Any number of agents $k\geq2$ cannot explore the given graph with a competitive ratio lower than $1.5$. The dotted line is a path consisting entirely of edges with length $\varepsilon$. Case 1 (Left): the graph an adversary reveals if $(s, c_3)$ is not traversed after an agent traversed a distance of $(J-3)\varepsilon$ on the path between $s$ and $c_2$. Case 2 (Right): the graph an adversary creates when $(s,c_3)$ is traversed before an agent traverses a distance of $j\varepsilon<(J-3)\varepsilon$ on the path between $s$ and $c_2$.
• Figure 5: Lower Bound construction for the energy model.

Theorems & Definitions (26)

• Theorem 3.1: Lower bound of ALE
• proof
• Theorem 3.2: ALE on tadpole graphs
• Theorem 4.1: The ALE algorithm and the energy model
• proof
• Theorem 4.2: Cycles in the Energy Model
• proof
• Theorem 5.1: Time Model: Lower bound for exploring tadpole graphs
• proof
• Theorem 5.2: Energy Model: Lower bound for exploring tadpole graphs