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.

翻译：本文研究了虚拟网络嵌入问题（VNE）的复杂性，该问题是与虚拟化技术（如网络切片）实施相关的若干电信问题的组合核心。VNE旨在寻找虚拟需求到物理资源的最优分配方案，同时包含放置与路由决策。已知该问题即使在虚拟网络为均匀路径时也是强NP难的，但在某些实际场景中具有多项式复杂度。本文旨在为VNE问题划定易解实例与难解实例之间的清晰边界。为此，我们考虑均匀需求以聚焦于结构特性而非封装特性。基于此，针对实践中出现的虚拟与物理网络特定拓扑结构（如树、环、轮图和团）展开研究。当物理网络为树或环结构时，提出了多项式时间的贪心或动态规划算法；而其他相近情形则被证明是NP难的。