Exploiting Automorphisms of Temporal Graphs for Fast Exploration and Rendezvous

TL;DR

This work studies exploration and rendezvous on connected temporal graphs through the lens of automorphisms and orbit structure. It introduces the orbit number $r$ as a structural parameter and proves deterministic upper bounds $O(r n^{1+ε})$ for TEXP and $O(n^{1+ε})$ for TRP in the clairvoyant setting, improving on general worst-case bounds when $r$ is sublinear. The paper also establishes lower bounds $Ω(n\log n)$ for TRP on single-orbit graphs and $Ω(rn)$ for TEXP across $r$-orbit graphs, highlighting near-tightness up to $n^{ε}$ factors. A randomized method for constructing a temporal walk visiting an entire orbit with probability at least $1-ε$ achieves $O((n^{5/3}+rn)\log n)$ time using $O(n^{1/3}\log(n/ε))$ scans, and an expectation-based variant avoids dependence on the automorphism group size. Overall, the results demonstrate that graph symmetries can yield substantially faster exploration and rendezvous in temporal networks and open paths for extending to multi-agent and symmetric rendezvous scenarios.

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 $O(r n^{1+ε})$ time steps, for any fixed $ε>0$. For $r=O(n^c)$ for constant $c<1$, this is a significant improvement over the known tight worst-case bound of $Θ(n^2)$ time steps for arbitrary connected temporal graphs. We also give two lower bounds for exploration, showing that $Ω(n \log n)$ time steps are required for some inputs with $r=O(1)$ and that $Ω(rn)$ 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 $O(n^{1+ε})$ time steps, for any $ε>0$. For some connected temporal graphs with constant orbit number we present a complementary lower bound of $Ω(n\log n)$ 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 $1-ε$ for any $0<ε<1$ such that $W$ spans $O((n^{5/3}+rn)\log n)$ time steps. The runtime of this algorithm consists of $O(n^{1/3} \log (n/ε))$ linear-time scans of the snapshots that exist in this time span.

Figures (5)

• Figure 1: Temporal graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ have different groups $\text{Aut}(\mathcal{G}_1)$ and $\text{Aut}(\mathcal{G}_2)$. $\text{Aut}(\mathcal{G}_1)$ has only the identity automorphism, i.e., $(u_1,u_2,u_3) \rightarrow (u_1,u_2,u_3)$ whereas $\text{Aut}(\mathcal{G}_2)$ has the identity automorphism and additionally the automorphism $(u_1,u_2,u_3) \rightarrow (u_3,u_2,u_1)$. This highlights the fact that temporal graphs with the same underlying graphs may have different automorphisms since the set of automorphisms of the temporal graph is the intersection of the set of automorphisms of the constituent graphs at each time step.
• Figure 2: Visualization of the properties of a lane $L_{t, t+r}(X)$. Each vertex in the set $X$ is colored black, and the set of reachable vertices in each time step is colored gray. Each figure represents one additional time step. In the second time step (Figure b) all vertices of the lane are reachable. In the third time step (Figure c) one vertex outside the lane must be reachable, for example the diamond shaped vertex. This is due to the fact that the temporal graph at hand is connected, and thus at least one additional vertex is reachable with every next time step. Here, one additional time step then suffices to reach a vertex of $S_2 \setminus X$ (Figure d).
• Figure 3: The construction scheme of Theorem \ref{['thm:orbitexplore']}. $W$ is the temporal walk constructed so far. We aim to extend the walk with a walk $W_{\sigma}$ that visits many vertices of $S\setminus T$, where $T$ is the set of vertices of orbit $S$ that we have already visited. To find $W_{\sigma}$, we construct $W'$ and the automorphism matrix (bottom left) for the vertex with label $10$ (the start vertex of $W'$), the set $Y$ of vertices of $S$ visited by $W'$, and the entire orbit $S$ as the set of possible start vertices of $W_{\sigma}$. One of the rows in the matrix then gives us an automorphism $\sigma$ that, when applied to $W'$, yields the desired walk $W_{\sigma}$.
• Figure 4: The initial cycle of phase $1$, $C_0$, on the left. Initially, the agents are placed on the vertices marked with dashed circles and can move up to $K$ vertices away in phase $1$. In each other phase $i \geq 2$, the cycle $C_i$ is viewed as consisting of sections, as shown on the right. At the beginning of the phase, the agents are placed in vertices of two sections (shown grey) that are sufficiently far apart, and they can move up to $K$ vertices away from the boundary of their section before the next phase starts.
• Figure 5: The initial cycle of phase $1$, $C_0$, on the left. Initially, the agents are placed on the vertices marked with dashed circles and can move up to $K$ vertices away in phase $1$. In each other phase $i \geq 2$, the cycle $C_i$ is viewed as consisting of sections, as shown on the right. At the beginning of the phase, the agents are placed in vertices of two sections (shown grey) that are sufficiently far apart, and they can move up to $K$ vertices away from the boundary of their section before the next phase starts.

Theorems & Definitions (71)

• lemma 1: Reachability, ErlebachHK21
• lemma 2
• proof
• lemma 3
• proof
• corollary 1
• proof
• lemma 4
• proof
• lemma 5: Reachability between Orbits