Exploiting Automorphisms of Temporal Graphs for Fast Exploration and Rendezvous (2312.07140v2)

Abstract: Temporal graphs are graphs where the edge set can change in each time step, and the vertex set stays the same. Exploration of temporal graphs whose snapshot in each time step is a connected graph, called connected temporal graphs, has been widely studied. We extend the concept of graph automorphisms from static graphs to temporal graphs and show that symmetries enable faster exploration: We prove that a connected temporal graph with $n$ vertices and orbit number $r$ (i.e., $r$ is the number of automorphism orbits) can be explored in $O(r n^{1+\epsilon})$ time steps, for any fixed $\epsilon>0$. For $r=O(n^c)$ for constant $c<1$, this is a significant improvement over the known tight worst-case bound of $\Theta(n^2)$ time steps for arbitrary connected temporal graphs. We also give two lower bounds for exploration, showing that $\Omega(n \log n)$ time steps are required for some inputs with $r=O(1)$ and that $\Omega(rn)$ time steps are required for some inputs for any $r$ with $1\le r\le n$. The techniques we develop for fast exploration are used to derive the following result for rendezvous in connected temporal graphs: Two agents are placed by an adversary at arbitrary vertices and given full information about the temporal graph, except that they do not have consistent vertex labels. The agents can meet at a common vertex after $O(n^{1+\epsilon})$ time steps, for any $\epsilon>0$. For some connected temporal graphs with constant orbit number we present a complementary lower bound of $\Omega(n\log n)$ time steps. Finally, we give a randomized algorithm to construct a temporal walk $W$ that visits all vertices of a given orbit with probability at least $1-\epsilon$ for any $0<\epsilon<1$ such that $W$ spans $O((n^{5/3}+rn)\log n)$ time steps. The runtime of this algorithm consists of $O(n^{1/3} \log (n/\epsilon))$ linear-time scans of the snapshots that exist in this time span.