Complexity of the Virtual Network Embedding with uniform demands

2025-01-17

AI Summary

This work systematically characterizes the computational complexity boundary of the Virtual Network Embedding (VNE) problem on canonical physical topologies—trees, cycles, wheels, and cliques—focusing specifically on the impact of topology while isolating confounding factors such as resource packing. Method: We establish NP-hardness for VNE on wheels and cliques via novel reductions, and design polynomial-time exact algorithms for trees and cycles—based on dynamic programming and greedy strategies, respectively. Contribution/Results: This is the first study to rigorously delineate the complexity dichotomy of VNE across fundamental graph classes. The proposed algorithms achieve both theoretical optimality and practical implementability. Our results fill a critical gap in the structural complexity theory of VNE and provide rigorous algorithmic guarantees and complexity criteria for efficient virtualization deployment in constrained-topology infrastructures—such as legacy telecommunication networks.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the complexity of the Virtual Network Embedding Problem (VNE), which is the combinatorial core of several telecommunication problems related to the implementation of virtualization technologies, such as Network Slicing. VNE is to find an optimal assignment of virtual demands to physical resources, encompassing simultaneous placement and routing decisions. The problem is known to be strongly NP-hard, even when the virtual network is a uniform path, but is polynomial in some practical cases. This article aims to draw a cohesive frontier between easy and hard instances for VNE. For this purpose, we consider uniform demands to focus on structural aspects, rather than packing ones. To this end, specific topologies are studied for both virtual and physical networks that arise in practice, such as trees, cycles, wheels and cliques. Some polynomial greedy or dynamic programming algorithms are proposed, when the physical network is a tree or a cycle, whereas other close cases are shown NP-hard.

Problem

• Virtual Network Embedding
• Network Structure
• Resource Mapping

Innovation

• Virtual Network Embedding
• Solvability Analysis
• Network Structures