Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): the problem of finding a mapping of a request graph (describing a workload) onto a substrate graph (describing the physical infrastructure). Applications range from mapping testbeds, over the embedding of batch-processing tasks to the embedding of service function chains and come with different mapping restrictions for nodes and edges. The restrictions studied most often are node and edge capacities, node mapping, edge routing and latency restrictions. While the VNEP has been studied intensively, complexity results are only known for specific models and this paper provides a first comprehensive study of the computational complexity of the VNEP by systematically analyzing its hardness for any combination of the above stated mapping restrictions. For all studied variants the ${ {\mathcal {NP}}\text {-completeness}}$ of the respective decision problems is shown. Furthermore, ${ {\mathcal {NP}}\text {-completeness}}$ results for finding approximate embeddings, which may, e.g., violate capacity constraints by certain factors, are derived. Lastly, it is also shown that all these results pertain when restricting the request graphs to planar and degree-bounded graphs. While theoretic in nature, our results have severe practical implications. Firstly, any optimization variant of the VNEP is ${ {\mathcal {NP}}\text {-hard}}$ and cannot be approximated for any of the studied restrictions, unless ${ {\mathcal {P}}{\,=\,} {\mathcal {NP}}}$ . Secondly, we uncover structural hardness properties: the VNEP is ${ {\mathcal {NP}}\text {-hard}}$ and inapproximable even if, e.g., only node placement and edge routing restrictions are considered.

中文翻译：

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虚拟网络嵌入的难与难

云中的许多资源分配问题都可以描述为一个基本的虚拟网络嵌入问题（VNEP）：查找网络的映射问题。 请求图 （描述工作量）到 底物图（描述物理基础结构）。应用范围从映射测试平台到批处理任务的嵌入到服务功能链的嵌入，并且对节点和边缘具有不同的映射限制。研究最多的限制是节点和边缘容量，节点映射，边缘路由和等待时间限制。尽管已经对VNEP进行了深入研究，但复杂度结果仅针对特定模型才是已知的，并且本文通过针对上述映射限制的任意组合系统地分析其硬度，从而提供了VNEP的计算复杂度的首次综合研究。对于所有研究的变体 $ {{\\ mathcal {NP}} \ text {-completeness}} $ 显示了各个决策问题。此外， $ {{\\ mathcal {NP}} \ text {-completeness}} $ 得出用于寻找近似嵌入的结果，该近似嵌入可能例如由于某些因素而违反容量限制。最后，还表明，当将请求图限制为平面图和度界图时，所有这些结果都适用。尽管从理论上讲，我们的结果具有严重的实际意义。首先，任何 VNEP的优化变体是 $ {{\\ mathcal {NP}} \ text {-hard}} $ 除非有任何研究限制，否则不能近似估算，除非 $ {{\ mathcal {P}} {\，= \，} {\ mathcal {NP}}} $ 。其次，我们发现结构硬度特性：VNEP为 $ {{\\ mathcal {NP}} \ text {-hard}} $ 即使仅考虑节点放置和边缘路由限制，也无法实现。