# On Virtual Network Embedding: Paths and Cycles

## TL;DR

This paper addresses the virtual network embedding problem focusing on paths and cycles, proposing specialized algorithms that improve acceptance ratios and revenue in network virtualization.

## Contribution

It introduces novel algorithms for path and cycle embedding, leveraging knapsack problem techniques and auxiliary graphs, tailored for specialized network topologies.

## Key findings

- Path embedding is NP-hard and inapproximable.
- Proposed algorithms outperform generic methods in acceptance ratio.
- Algorithms increase revenue in network virtualization scenarios.

## Abstract

Network virtualization provides a promising solution to overcome the ossification of current networks, allowing multiple Virtual Network Requests (VNRs) embedded on a common infrastructure. The major challenge in network virtualization is the Virtual Network Embedding (VNE) problem, which is to embed VNRs onto a shared substrate network and known to be $\mathcal{NP}$-hard. The topological heterogeneity of VNRs is one important factor hampering the performance of the VNE. However, in many specialized applications and infrastructures, VNRs are of some common structural features $\textit{e.g.}$, paths and cycles. To achieve better outcomes, it is thus critical to design dedicated algorithms for these applications and infrastructures by taking into accounting topological characteristics. Besides, paths and cycles are two of the most fundamental topologies that all network structures consist of. Exploiting the characteristics of path and cycle embeddings is vital to tackle the general VNE problem. In this paper, we investigated the path and cycle embedding problems. For path embedding, we proved its $\mathcal{NP}$-hardness and inapproximability. Then, by utilizing Multiple Knapsack Problem (MKP) and Multi-Dimensional Knapsack Problem (MDKP), we proposed an efficient and effective MKP-MDKP-based algorithm. For cycle embedding, we proposed a Weighted Directed Auxiliary Graph (WDAG) to develop a polynomial-time algorithm to determine the least-resource-consuming embedding. Numerical results showed our customized algorithms can boost the acceptance ratio and revenue compared to generic embedding algorithms in the literature.