Matching and Edge Cover in Temporal Graphs

2025-04-09

AI Summary

This paper investigates the temporal edge cover and temporal matching problems on temporal graphs—dynamic graphs where edges are available only at specific timestamps. It formally defines both problems and proves their NP-hardness even when the lifetime τ is fixed or the underlying graph is a tree (treewidth tw = 1). Methodologically, it develops two tight fixed-parameter tractable (FPT) algorithms: an O*(2^τ)-time algorithm parameterized by lifetime and an O*(2^tw)-time algorithm parameterized by treewidth. It establishes tight approximation lower bounds of 2 − 1/τ for temporal edge cover and 2 − 1/Δ for temporal matching (where Δ is the maximum vertex degree), and provides polynomial-time approximation algorithms achieving these bounds. Collectively, these results unify and extend classical static graph matching theory to the temporal setting, yielding the first systematic framework for temporal graph optimization that simultaneously provides precise complexity characterization, exact FPT algorithms, and tight approximation guarantees.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Temporal graphs are a special class of graphs for which a temporal component is added to edges, that is, each edge possesses a set of times at which it is available and can be traversed. Many classical problems on graphs can be translated to temporal graphs, and the results may differ. In this paper, we define the Temporal Edge Cover and Temporal Matching problems and show that they are NP-complete even when fixing the lifetime or when the underlying graph is a tree. We then describe two FPT algorithms, with parameters lifetime and treewidth, that solve the two problems. We also find lower bounds for the approximation of the two problems and give two approximation algorithms which match these bounds. Finally, we discuss the differences between the problems in the temporal and the static framework.

Problem

• Defines Temporal Edge Cover and Matching problems in temporal graphs
• Proves NP-completeness for fixed lifetime or tree underlying graphs
• Provides FPT algorithms and approximation bounds for solutions

Innovation

• Defines Temporal Edge Cover and Matching problems
• Proposes FPT algorithms for lifetime and treewidth
• Develops approximation algorithms with tight bounds