Virtual networks are an innovative abstraction that extends cloud computing concepts to the network: by supporting bandwidth reservations between compute nodes (e.g., virtual machines), virtual networks can provide a predictable performance to distributed and communication-intensive cloud applications. However, in order to make the most efficient use of the shared resources, the Virtual Network Embedding (VNE) problem has to be solved: a virtual network should be mapped onto the given physical network so that resource reservations are minimized. The problem has been studied intensively already and is known to be NP-hard in general. In this paper, we revisit this problem and consider it on specific topologies, as they often arise in practice. To be more precise, we study the weighted version of the VNE problem: we consider a virtual weighted network of a specific topology which we want to embed onto a weighted network with capacities and specific topology. As for topologies, we consider most fundamental and commonly used ones: line, star, $2$-tiered star, oversubscribed $2$-tiered star, and tree, in addition to also considering arbitrary topologies. We show that typically the VNE problem is NP-hard even in more specialized cases, however, sometimes there exists a polynomial algorithm: for example, an embedding of the oversubscribed $2$-tiered star onto the tree is polynomial while an embedding of an arbitrary $2$-tiered star is not.

翻译：虚拟网络是一种创新的抽象概念，将云计算理念延伸至网络领域：通过支持计算节点（如虚拟机）间的带宽预留，虚拟网络可为分布式及通信密集型云应用提供可预测的性能保障。然而，为最大化共享资源利用效率，需解决虚拟网络嵌入问题：将虚拟网络映射至给定物理网络时需最小化资源预留。该问题已被广泛研究，且已知在一般情况下属于NP难问题。本文重新审视该问题，并针对实际场景中常见的特定拓扑展开研究。具体而言，我们研究加权版本的VNE问题：考虑将特定拓扑的加权虚拟网络嵌入至具有容量约束和特定拓扑的加权物理网络。在拓扑类型上，我们聚焦最基础且最常用的结构：线形、星形、双层星形、超量订阅双层星形、树形，同时涵盖任意拓扑的分析。研究结果表明，即便在更特殊的情形下，VNE问题通常仍为NP难问题；然而在部分场景中存在多项式时间算法：例如，超量订阅双层星形至树形拓扑的嵌入属于多项式可解问题，而任意双层星形拓扑的嵌入则不然。