Parameterized Complexity of Temporal Connected Components: Treewidth and k-Path Graphs

2025-10-07

AI Summary

This paper investigates the parameterized complexity of computing maximum temporal connected components (TCCs) in temporal graphs, focusing on the NP-hard openTCC and closedTCC variants. Methodologically, it introduces the *temporal path number* (τ)—the minimum number of temporal paths covering all edges—as a novel structural parameter, and systematically analyzes its interplay with classical parameters such as treewidth (tw) and maximum degree (Δ). Theoretically, it establishes that openTCC is XP with respect to τ alone, while both TCC variants become fixed-parameter tractable (FPT) under combined parameters (τ + tw), (τ + Δ), and (tw + Δ); neither τ nor tw alone suffices for FPT. The approach integrates temporal path modeling, tree decompositions, and dynamic programming to precisely delineate the frontier of parameterized tractability. These results uncover a new paradigm wherein temporal structure (captured by τ) and static topology (e.g., tw) jointly govern connectivity, advancing the theoretical foundations of temporal graph algorithms.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We study the parameterized complexity of maximum temporal connected components (tccs) in temporal graphs, i.e., graphs that deterministically change over time. In a tcc, any pair of vertices must be able to reach each other via a time-respecting path. We consider both problems of maximum open tccs (openTCC), which allow temporal paths through vertices outside the component, and closed tccs (closedTCC) which require at least one temporal path entirely within the component for every pair. We focus on the structural parameter of treewidth, tw, and the recently introduced temporal parameter of temporal path number, tpn, which is the minimum number of paths needed to fully describe a temporal graph. We prove that these parameters on their own are not sufficient for fixed parameter tractability: both openTCC and closedTCC are NP-hard even when tw=9, and closedTCC is NP-hard when tpn=6. In contrast, we prove that openTCC is in XP when parameterized by tpn. On the positive side, we show that both problem become fixed parameter tractable under various combinations of structural and temporal parameters that include, tw plus tpn, tw plus the lifetime of the graph, and tw plus the maximum temporal degree.

Problem

• Analyzing maximum temporal connected components in dynamic graphs over time
• Studying parameterized complexity using treewidth and temporal path parameters
• Determining fixed parameter tractability for open and closed temporal components

Innovation

• Combining treewidth and temporal path number for FPT
• Using treewidth plus lifetime for fixed parameter tractability
• Employing treewidth plus temporal degree for FPT