Complexity of the Virtual Network Embedding with uniform demands

TL;DR

This paper investigates the complexity frontier of the Virtual Network Embedding Problem under uniform demands (uniVNE), formalizing the problem with the $\alpha$-$\langle \beta_r \rightarrow \gamma_s \rangle$ taxonomy to compare embeddings across virtual/topology and substrate/topology combinations. It leverages classical computational problems (e.g., Hamiltonian Graph Problem, Hadwinger Number Problem, EDPP) to prove NP-hardness for many configurations (notably cycles, wheels, and cliques on general substrates) while delivering polynomial-time dynamic programming or greedy algorithms for several practical topologies on substrates such as trees, cycles, and cliques. The results collectively map a rich landscape: easy instances exist for specific substrate/topology pairs (e.g., tree substrates with many virtual topologies, certain cycle/path/clique cases on cycles or trees), whereas other combinations remain NP-hard or open. The findings offer guidance for designing VNE solvers and optimization formulations in real networks (e.g., data centers and fiber networks) and lay out open directions, including extending DP/flow-based techniques to multi-virtual-graph embeddings and exploring polyhedral approaches for the proven polynomial cases. Overall, the work advances understanding of when uniVNE can be solved efficiently and where inherent combinatorial difficulty remains, with direct implications for network virtualization and slicing in modern telecommunications.

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.

Figures (4)

• Figure 1: Example of a mapping
• Figure 2: Counter example for the reduction from HNP to E-$\langle \mathcal{K}_r \rightarrow \mathcal{G}_s \rangle$ by tieves2016complexity
• Figure 3: Example of a node placement swap from $a)$ to $b)$ from proof of Lemma \ref{['lemma:section4:cycle:consec']}
• Figure 4: Example of a mapping of $\mathcal{K}_4$ on $\mathcal{C}_5$

Theorems & Definitions (41)

• Theorem 1
• proof
• Corollary 2
• proof
• Theorem 3
• proof
• Proposition 4
• proof
• Theorem 5
• proof