Complexity of the Virtual Network Embedding with uniform demands

TL;DR

This paper investigates the computational complexity of the Virtual Network Embedding problem with uniform demands, identifying which network topologies allow polynomial solutions and which are NP-hard, thus clarifying the problem's difficulty landscape.

Contribution

It provides a systematic analysis of the VNE problem's complexity for various network topologies with uniform demands, proposing algorithms for some cases and proving hardness for others.

Findings

Polynomial algorithms for trees and cycles

NP-hardness results for certain topologies

Clarification of easy vs. hard instances based on topology

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.