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_7: We consider the task of embedding multiple service requests in Software-Defined Networks (SDNs), i.e. computing (combined) mappings of network functions on physical nodes and finding routes to connect the mapped network functions. A single service request may either be fully embedded or be rejected. The objective is to maximize the sum of benefits of the served requests, while the solution must abide node and edge capacities.
  @cite_8: It is well-known that cloud application performance can critically depend on the network. Over the last years, several systems have been developed which provide the application with the illusion of a virtual cluster: a star-shaped virtual network topology connecting virtual machines to a logical switch with absolute bandwidth guarantees. In this paper, we debunk some of the myths around the virtual cluster embedding problem. First, we show that the virtual cluster embedding problem is not NP-hard, and present the fast and optimal embedding algorithm VC-ACE for arbitrary datacenter topologies. Second, we argue that resources may be wasted by enforcing star-topology embeddings, and alternatively promote a hose embedding approach. We discuss the computational complexity of hose embeddings and derive the HVC-ACE algorithm. Using simulations we substantiate the benefits of hose embeddings in terms of acceptance ratio and resource footprint.
  @cite_9: 
  @cite_0: The shared nature of the network in today's multi-tenant datacenters implies that network performance for tenants can vary significantly. This applies to both production datacenters and cloud environments. Network performance variability hurts application performance which makes tenant costs unpredictable and causes provider revenue loss. Motivated by these factors, this paper makes the case for extending the tenant-provider interface to explicitly account for the network. We argue this can be achieved by providing tenants with a virtual network connecting their compute instances. To this effect, the key contribution of this paper is the design of virtual network abstractions that capture the trade-off between the performance guarantees offered to tenants, their costs and the provider revenue. To illustrate the feasibility of virtual networks, we develop Oktopus, a system that implements the proposed abstractions. Using realistic, large-scale simulations and an Oktopus deployment on a 25-node two-tier testbed, we demonstrate that the use of virtual networks yields significantly better and more predictable tenant performance. Further, using a simple pricing model, we find that the our abstractions can reduce tenant costs by up to 74 while maintaining provider revenue neutrality.
  @cite_15: The virtualization and softwarization of modern computer networks enables the definition and fast deployment of novel network services called service chains: sequences of virtualized network functions e.g., firewalls, caches, traffic optimizers through which traffic is routed between source and destination. This paper attends to the problem of admitting and embedding a maximum number of service chains, i.e., a maximum number of source-destination pairs which are routed via a sequence of l to-be-allocated, capacitated network functions. We consider an Online variant of this maximum Service Chain Embedding Problem, short OSCEP, where requests arrive over time, in a worst-case manner. Our main contribution is a deterministic Ologl-competitive online algorithm, under the assumption that capacities are at least logarithmic in l. We show that this is asymptotically optimal within the class of deterministic and randomized online algorithms. We also explore lower bounds for offline approximation algorithms, and prove that the offline problem is APX-hard for unit capacities and smalli¾źli¾ź3, and even Poly-APX-hard in general, when there is no bound oni¾źl. These approximation lower bounds may be of independent interest, as they also extend to other problems such as Virtual Circuit Routing. Finally, we present an exact algorithm based on 0-1 programming, implying that the general offline SCEP is in NP and, by the above hardness results, it is NP-complete for constant l.
  @cite_11: We study a basic resource allocation problem that arises in cloud computing environments. The physical network of the cloud is represented as a graph with vertices denoting servers and edges corresponding to communication links. A workload is a set of processes with processing requirements and mutual communication requirements. The workloads arrive and depart over time, and the resource allocator must map each workload upon arrival to the physical network. We consider the objective of minimizing the congestion. We show that solving a subproblem about mapping a single workload to the physical graph essentially suffices to solve the general problem. In particular, an α-approximation for this single mapping problem gives an O(α log nD)-competitive algorithm for the general problem, where n is the number of nodes in the physical network and D is the maximum to minimum workload duration ratio. We also show how to solve the single mapping problem for two natural class of workloads, namely depth-d-trees and complete-graph workloads. For depth-d tree, we give an nO(d) time O(d2 log (nd))-approximation based on a strong LP relaxation inspired by the Sherali-Adams hierarchy.
summary: Acknowledging the hardness of the general VNEP and the diversity of applications, several subproblems of the VNEP have been studied recently by considering restricted graph classes for the virtual networks and the substrate graph. For example, virtual clusters with uniform demands are studied in @cite_0 @cite_8 , line requests are studied in @cite_9 @cite_7 @cite_15 and tree requests were studied in @cite_7 @cite_11 .