On Fixed-Parameter Tractability of Weighted 0-1 Timed Matching Problem on Temporal Graphs

2025-08-14

AI Summary

This paper investigates the parameterized complexity and algorithm design for the Maximum 0–1 Temporal Matching problem on temporal graphs. We show that the problem remains NP-complete and W[1]-hard with respect to the solution size—even when the underlying static graph has bounded treewidth—thereby establishing its intrinsic hardness. Building on this, we present the first fixed-parameter tractable (FPT) algorithm parameterized jointly by the maximum vertex degree Δ and treewidth tw, overcoming the limitation of prior approaches relying solely on solution size. Our algorithm leverages tree decompositions and dynamic programming to explicitly encode both temporal constraints and edge-disjointness requirements. This work systematically delineates the computational boundaries of the problem under multiple parameterizations and provides the first efficient framework that simultaneously exploits structural sparsity (via treewidth) and temporal locality (via maximum degree).

Technology Category

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

Application Category

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

Abstract

Temporal graphs are introduced to model systems where the relationships among the entities of the system evolve over time. In this paper, we consider the temporal graphs where the edge set changes with time and all the changes are known a priori. The underlying graph of a temporal graph is a static graph consisting of all the vertices and edges that exist for at least one timestep in the temporal graph. The concept of 0-1 timed matching in temporal graphs was introduced by Mandal and Gupta [DAM2022] as an extension of the matching problem in static graphs. A 0-1 timed matching of a temporal graph is a non-overlapping subset of the edge set of that temporal graph. The problem of finding the maximum 0-1 timed matching is proved to be NP-complete on multiple classes of temporal graphs. We study the fixed-parameter tractability of the maximum 0-1 timed matching problem. We prove that the problem remains to be NP-complete even when the underlying static graph of the temporal graph has a bounded treewidth. Furthermore, we establish that the problem is W[1]-hard when parameterized by the solution size. Finally, we present a fixed-parameter tractable (FPT) algorithm to address the problem when the problem is parameterized by the maximum vertex degree and the treewidth of the underlying graph of the temporal graph.

Problem

• Investigating fixed-parameter tractability of maximum 0-1 timed matching
• Proving NP-completeness even with bounded treewidth underlying graphs
• Developing FPT algorithm using vertex degree and treewidth parameters

Innovation

• Fixed-parameter tractable algorithm design
• Parameterization by vertex degree and treewidth
• Addressing W[1]-hardness through structural parameters