Improved exploration of temporal graphs

2025-11-27

AI Summary

This paper studies the Shortest Exploration Sequence problem on temporal graphs: given a connected temporal graph sequence $(G_t)_{t in I}$ with bounded maximum degree, find a walk starting from a designated vertex—where at each step the agent may either wait or traverse an edge present at the current time—that visits all vertices in minimum steps. We present the first general upper bound of $O(n^{3/2}sqrt{D log n})$ on the exploration length, where $D$ is the average temporal maximum degree. When $D = o(n / log n)$, this yields subquadratic exploration time, improving upon the prior best bound of $O(n^{7/4})$. Our approach integrates connectivity constraints, analysis of vertex-degree distribution over time, and a dynamic adaptation mechanism. The framework applies uniformly to structured graph families—including planar graphs and graphs of bounded treewidth—and establishes the first subquadratic exploration guarantee under a bounded average-degree assumption.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

A temporal graph $G$ is a sequence $(G_t)_{t in I}$ of graphs on the same vertex set of size $n$. The emph{temporal exploration problem} asks for the length of the shortest sequence of vertices that starts at a given vertex, visits every vertex, and at each time step $t$ either stays at the current vertex or moves to an adjacent vertex in $G_t$. Bounds on the length of a shortest temporal exploration have been investigated extensively. Perhaps the most fundamental case is when each graph $G_t$ is connected and has bounded maximum degree. In this setting, Erlebach, Kammer, Luo, Sajenko, and Spooner [ICALP 2019] showed that there exists an exploration of $G$ in $mathcal{O}(n^{7/4})$ time steps. We significantly improve this bound by showing that $mathcal{O}(n^{3/2} sqrt{log n})$ time steps suffice. In fact, we deduce this result from a much more general statement. Let the emph{average temporal maximum degree} $D$ of $G$ be the average of $max_{t in I} d_{G_t}(v)$ over all vertices $v in V(G)$, where $d_{G_t}(v)$ denotes the degree of $v$ in $G_t$. If each graph $G_t$ is connected, we show that there exists an exploration of $G$ in $mathcal{O}(n^{3/2} sqrt{D log n})$ time steps. In particular, this gives the first subquadratic upper bound when the underlying graph has bounded average degree. As a special case, this also improves the previous best bounds when the underlying graph is planar or has bounded treewidth and provides a unified approach for all of these settings. Our bound is subquadratic already when $D=o(n/log n)$.

Problem

• Improves upper bound for temporal graph exploration time
• Introduces average temporal maximum degree for analysis
• Provides subquadratic bounds for connected bounded-degree graphs

Innovation

• Improved bound to O(n^{3/2} sqrt(log n)) steps
• Introduces average temporal maximum degree D
• General bound O(n^{3/2} sqrt(D log n))