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_18: 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 (from where the problem originated), over the embedding of batch-processing workloads (virtual clusters) to the embedding of service function chains. The different applications come with their own specific requirements and constraints, including node mapping constraints, routing policies, and latency constraints. While the VNEP has been studied intensively over the last years, complexity results are only known for specific models and we lack a comprehensive understanding of its hardness.This paper charts the complexity landscape of the VNEP by providing a systematic analysis of the hardness of a wide range of VNEP variants, using a unifying and rigorous proof framework. In particular, we show that the problem of finding a feasible embedding is @math -complete in general, and, hence, the VNEP cannot be approximated under any objective, unless @math holds. Importantly, we derive @math -completeness results also for finding approximate embeddings, which may violate, e.g., capacity constraints by certain factors. Lastly, we prove that our results still pertain when restricting the request graphs to planar or degree-bounded graphs.
  @cite_1: Recently network virtualization has been proposed as a promising way to overcome the current ossification of the Internet by allowing multiple heterogeneous virtual networks (VNs) to coexist on a shared infrastructure. A major challenge in this respect is the VN embedding problem that deals with efficient mapping of virtual nodes and virtual links onto the substrate network resources. Since this problem is known to be NP-hard, previous research focused on designing heuristic-based algorithms which had clear separation between the node mapping and the link mapping phases. This paper proposes VN embedding algorithms with better coordination between the two phases. We formulate the VN em- bedding problem as a mixed integer program through substrate network augmentation. We then relax the integer constraints to obtain a linear program, and devise two VN embedding algo- rithms D-ViNE and R-ViNE using deterministic and randomized rounding techniques, respectively. Simulation experiments show that the proposed algorithms increase the acceptance ratio and the revenue while decreasing the cost incurred by the substrate network in the long run.
  @cite_6: Network virtualization is recognized as an enabling technology for the future Internet. It aims to overcome the resistance of the current Internet to architectural change. Application of this technology relies on algorithms that can instantiate virtualized networks on a substrate infrastructure, optimizing the layout for service-relevant metrics. This class of algorithms is commonly known as "Virtual Network Embedding (VNE)" algorithms. This paper presents a survey of current research in the VNE area. Based upon a novel classification scheme for VNE algorithms a taxonomy of current approaches to the VNE problem is provided and opportunities for further research are discussed.
  @cite_2: Network virtualization is a promising new resource management approach that allows customized virtual networks (VNs) to be multiplexed on a shared physical infrastructure. In this paper, our focus is on the embedding of VN resources onto this infrastructure. Since this problem is known to be NP-hard, embedding proposals in literature are heuristic-based approaches that restrict the problem space in different dimensions. Limitations of these proposals are: 1) as embedding of VN links and nodes is performed in two separate stages, it may ensue in a high blocking of VN requests and a less efficient usage of substrate resources; and 2) as pricing of embedding resources is based on linear functions, it triggers no competition among VN users in order to maximize infrastructure provider profits. These drawbacks motivate us to propose a mathematical model that makes use of large-scale optimization tools and proposes a Column Generation (CG) formulation of the problem, coupled with branch-and-bound technique or rounding-off heuristic. We also propose a periodical planning of embedding process where profitable VN requests are selected through an auction mechanism. In our experiments with different substrate network topologies and many different VN request patterns, we show a clear advantage of auction-based CG models over present benchmarks .
  @cite_13: Distributed applications often require high-performance networks with strict connectivity guarantees. For instance, many cloud applications suffer from today's variations of the intra-cloud bandwidth, which leads to poor and unpredictable application performance. Accordingly, we witness a trend towards virtual networks (VNets) which can provide resource isolation. Interestingly, while the problem of where to embed a VNet is fairly well-understood today, much less is known about when to optimally allocate a VNet. This however is important, as the requirements specified for a VNet do not have to be static, but can vary over time and even include certain temporal flexibilities. This paper initiates the study of the temporal VNet embedding problem (TVNEP). We propose a continuous-time mathematical programming approach to solve the TVNEP, and present and compare different algorithms. Based on these insights, we present the CSM-Model which incorporates both symmetry and state-space reductions to significantly speed up the process of computing exact solutions to the TVNEP. Based on the CSM-Model, we derive a greedy algorithm OGA to compute fast approximate solutions. In an extensive computational evaluation, we show that despite the hardness of the TVNEP, the CSM-Model is sufficiently powerful to solve moderately sized instances to optimality within one hour and under different objective functions (such as maximizing the number of embeddable VNets). We also show that the greedy algorithm exploits flexibilities well and yields good solutions. More generally, our results suggest that already little time flexibilities can improve the overall system performance significantly.
  @cite_20: SDN simplifies network management by relying on declarativity (high-level interface) and expressiveness (network flexibility). We propose a solution to support those features while preserving high robustness and scalability as needed in carrier-grade networks. Our solution is based on (i) a two-layer architecture separating connectivity and optimization tasks; and (ii) a centralized optimizer called framework, which translates high-level goals expressed almost in natural language into compliant network configurations. Our evaluation on real and synthetic topologies shows that framework improves the state of the art by (i) achieving better trade-offs for classic goals covered by previous works, (ii) supporting a larger set of goals (refined traffic engineering and service chaining), and (iii) optimizing large ISP networks in few seconds. We also quantify the gains of our implementation, running Segment Routing on top of IS-IS, over possible alternatives (RSVP-TE and OpenFlow).
summary: In the last decade, the VNEP has attracted much attention due to its many applications and the survey @cite_6 from 2013 already lists more than 80 different algorithms for its many variations @cite_6 . The VNEP is known to be @math -hard and inapproximable in general (unless @math ) @cite_18 . Based on the hardness of the VNEP, most works consider heuristics without any performance guarantee @cite_6 @cite_1 . Other works proposed exact methods as integer or constraint programming, coming at the cost of an exponential runtime @cite_20 @cite_2 @cite_13 .