text: The Virtual Network Embedding Problem (VNEP) captures the essence of many resource allocation problems of today’s infrastructure 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 which specifies computational requirements at the nodes and communication requirements on the edges. An embedding of a Virtual Network on the shared physical infrastructure is the joint mapping of (virtual) nodes to physical servers together with the mapping of (virtual) edges onto paths in the physical network connecting the respective servers.This work initiates the study of approximation algorithms for the VNEP. Concretely, we study the offline setting with admission control: given multiple request graphs the task is to embed the most profitable subset while not exceeding resource capacities. Our approximation is based on the randomized rounding of Linear Programming (LP) solutions. Interestingly, we uncover that the standard LP formulation exhibits an inherent structural deficit when considering general virtual networks: its solutions cannot be decomposed into valid embeddings. In turn, focusing on the class of cactus request graphs, we devise a novel LP formulation, whose solutions can be decomposed into convex combinations of valid embedding. Proving performance guarantees of our rounding scheme, we obtain the first approximation algorithm for the VNEP in the resource augmentation model.We propose two rounding heuristics and evaluate their performance in an extensive computational study, showing that these consistently yield good solutions (even without augmentations).
  @cite_5: The SDN and NFV paradigms enable novel network services which can be realized and embedded in a flexible and rapid manner. For example, SDN can be used to flexibly steer traffic from a source to a destination through a sequence of virtualized middleboxes, in order to realize so-called service chains. The service chain embedding problem consists of three tasks: admission control, finding suitable locations to allocate the virtualized middleboxes and computing corresponding routing paths. This paper considers the offline batch embedding of multiple service chains. Concretely, we consider the objectives of maximizing the profit by embedding an optimal subset of requests or minimizing the costs when all requests need to be embedded. Interestingly, while the service chain embedding problem has recently received much attention, so far, only non- polynomial time algorithms (based on integer programming) as well as heuristics (which do not provide any formal guarantees) are known. This paper presents the first polynomial time service chain approximation algorithms both for the case with admission and without admission control. Our algorithm is based on a novel extension of the classic linear programming and randomized rounding technique, which may be of independent interest. In particular, we show that our approach can also be extended to more complex service graphs, containing cycles or sub-chains, hence also providing new insights into the classic virtual network embedding problem.
  @cite_10: Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.
summary: In our preliminary technical report @cite_5 similar results were presented. The current work presents a significantly simpler LP formulation and also provides an extensive computational evaluation. Additionally, in our recent technical report @cite_10 , the approximation approach presented in this work is extended beyond cactus request graphs. However, approximating more general request graphs comes at the price of non-polynomial runtimes.