Temporal Graph Reconfiguration for Always-Connected Graphs

TL;DR

The paper studies reconfiguring temporal graphs from $\mathcal{G}_1$ to $\mathcal{G}_2$ while every snapshot $\mathcal{G}(t)$ remains connected, introducing the Layered Connectivity Reconfiguration problem. It develops the reachability partition framework for bridges, enabling a polynomial-time algorithm that either constructs a reconfiguration sequence of length at most $2M^2$ or proves impossibility, with overall runtime $O(M^3)$. It further shows that finding the shortest reconfiguration sequence is NP-hard, via a Vertex Cover reduction even for lifetime $T=2$, highlighting fundamental limits. The results lay groundwork for temporal-graph reconfiguration, suggesting future work on approximation, restricted graph classes, and parameterized analyses.

Abstract

Network redesign problems ask to modify the edges of a given graph to satisfy some properties. In temporal graphs, where edges are only active at certain times, we are sometimes only allowed to modify when the edges are going to be active. In practice, we might not even be able to perform all of the necessary modifications at once; changes must be applied step-by-step while the network is still in operation, meaning that the network must continue to satisfy some properties. To initiate a study in this area, we introduce the temporal graph reconfiguration problem. As a starting point, we consider the Layered Connectivity Reconfiguration problem in which every snapshot of the temporal graph must remain connected throughout the reconfiguration. We provide insights into how bridges can be reconfigured into non-bridges based on their reachability partitions, which lets us identify any edge as either changeable or unchangeable. From this we construct a polynomial-time algorithm that gives a valid reconfiguration sequence of length at most 2M^2 (where M is the number of temporal edges), or determines that reconfiguration is not possible. We also show that minimizing the length of the reconfiguration sequence is NP-hard via a reduction from vertex cover.

Figures (8)

• Figure 1: Example of a valid reconfiguration sequence $(\mathcal{G}^0, \cdots, \mathcal{G}^4)$ between $\mathcal{G}_1$ and $\mathcal{G}_2$. Time labels are shown on each edge, and edges marked in blue differ from the previous graph in the sequence.
• Figure 2: Example with no valid reconfiguration sequence between $\mathcal{G}_1$ and $\mathcal{G}_2$, as changing any label in $\mathcal{G}_1$ disconnects a snapshot.
• Figure 3: Example of temporal graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ in which all different edges in $\mathcal{G}_1$ (marked in orange) are bridges.
• Figure 4: Example of the reachability partition of the edge $(e,2) = (\{C, D\},2)$.
• Figure 5: Illustration of $k$-changeable edges in a temporal graph. The $0$-changeable edges are exactly the non-bridges. For $i\in \mathbb{N}$, the $(i+1)$-changeable edges are those bridges that become non-bridges by changing an $i$-changeable edge.

Theorems & Definitions (11)

• Definition 0: Reachability Partition
• Lemma 0: Bridges and Reachability Partition
• Lemma 0: Reachability Partition invariant
• Theorem 1: Finding a reconfiguration sequence in polynomial-time
• Definition 1: $k$-Changeable Edge
• Lemma 1: Reachability Partition and $k$-changeability
• Lemma 1: Continuity of Changeable Edges
• Lemma 1: $\Change$ is computable in Polynomial-Time
• Lemma 1: Sequence for decreasing difference
• Theorem 1: Finding a reconfiguration sequence in polynomial-time