Temporal Graph Reconfiguration for Always-Connected Graphs

2025-10-17

AI Summary

This paper studies the dynamic reconfiguration of edge activity intervals in temporal graphs, aiming to iteratively adjust temporal edges while preserving connectivity of every snapshot. We introduce the first formal model for connectivity-preserving temporal graph reconfiguration, grounded in reachability-based partitioning to identify bridge edges whose temporal intervals can be safely modified. Methodologically, we integrate hierarchical connectivity modeling, structural analysis of bridge edges, and time-layered snapshot techniques to design a polynomial-time algorithm that constructs a valid reconfiguration sequence of length at most $2M^2$, where $M$ is the number of temporal edges. We also prove that minimizing sequence length is NP-hard. Our core contributions are: (i) the first formalization of this reconfiguration problem; (ii) a rigorous theoretical framework establishing necessary and sufficient conditions for safe reconfiguration; and (iii) an efficient algorithm with provable guarantees alongside a complexity-theoretic characterization.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Modifying temporal edge activations while maintaining connectivity
• Ensuring network connectivity during step-by-step reconfiguration process
• Finding valid reconfiguration sequences for temporal graph bridges

Innovation

• Reconfigures temporal edges while maintaining connectivity
• Classifies edges as changeable using reachability partitions
• Provides polynomial-time algorithm for valid reconfiguration sequences