Virtual Network Embedding Approximations: Leveraging Randomized Rounding
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The Virtual Network Embedding Problem (VNEP) captures the essence of many resource allocation problems of today's Cloud Providers, which offer their physical computation and networking resources to customers. Customers request resources in the form of Virtual Networks, i.e. as a directed graph, specifying computational requirements at the nodes and bandwidth requirements on the edges. An embedding of a Virtual Network on the shared physical infrastructure is the joint mapping of (virtual) nodes to suitable physical servers together with the mapping of (virtual) edges onto paths in the physical network connecting the respective servers. We study the offline setting of the VNEP in which multiple requests are given and the task is to find the most profitable set of requests to embed while not exceeding the physical resource capacities. This paper initiates the study of approximation algorithms for the VNEP by employing randomized rounding of Linear Programming solutions. We show that the standard Linear Programming formulation exhibits an inherent structural deficit, yielding large (or even infinite) integrality gaps. In turn, focusing on the class of cactus graphs for virtual networks, we devise a novel Linear Programming formulation together with an algorithm to decompose fractional solutions into convex combinations of valid embeddings. Applying randomized rounding, we obtain the first tri-criteria approximation algorithm in the classic resource augmentation model.
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