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).
A column generation approach was proposed by in @cite_2 to efficiently compute solutions to the VNEP by generating feasible mappings on-the-fly subject to a specific cost measure. In particular, compute feasible mappings by relying on heuristics and, if no feasible heuristical solution was found, rely on Mixed-Integer Programming to compute a cost optimal feasible embedding in non-polynomial time. We believe that our work can bridge the gap between the heuristic generation of feasible mappings and the optimal generation of feasible mappings as our approach can be used to compute mappings.
@cite_2
2186282916 2059695025 2792492052 2112788936
This paper studies a routing problem in a multimodal network with shipment consolidation options. A freight forwarder can use a mix of flexible-time and scheduled transportation services. Time windows are a prominent aspect of the problem. For instance, they are used to model opening hours of the terminals, as well as pickup and delivery time slots. The various features of the problem can be described as elements of a digraph and their integration leads to a holistic graph representation. This allows an origin- destination integer multi-commodity flow formulation with non-convex piecewise linear costs, time windows, and side constraints. Column generation algorithms are designed to compute lower bounds. These column generation algorithms are also embedded within heuristics aimed at finding feasible integer solutions. Computational results with real-life data are presented and show the efficacy of the proposed approach. This paper presents a heuristic method based on column generation for the EDF (Electricite De France) long-term electricity production planning problem proposed as subject of the ROADEF EURO 2010 Challenge. This is to our knowledge the first-ranked method among those methods based on mathematical programming, and was ranked fourth overall. The problem consists in determining a production plan over the whole time horizon for each thermal power plant of the French electricity company, and for nuclear plants, a schedule of plant outages which are necessary for refueling and maintenance operations. The average cost of the overall outage and production planning, computed over a set of demand scenarios, is to be minimized. The method proceeds in two stages. In the first stage, dates for outages are fixed once for all for each nuclear plant. Data are aggregated with a single average scenario and reduced time steps, and a set-partitioning reformulation of this aggregated problem is solved for fixing outage dates with a heuristic based on column generation. The pricing problem associated with each nuclear plant is a shortest path problem in an appropriately constructed graph. In the second stage, the reload level is determined at each date of an outage, considering now all scenarios. Finally, the production quantities between two outages are optimized for each plant and each scenario by solving independent linear programming problems. 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. We consider packing LP's with m rows where all constraint coefficients are normalized to be in the unit interval. The n columns arrive in random order and the goal is to set the corresponding decision variables irrevocably when they arrive to obtain a feasible solution maximizing the expected reward. Previous (1−e)-competitive algorithms require the right-hand side of the LP to be @math , a bound that worsens with the number of columns and rows. However, the dependence on the number of columns is not required in the single-row case and known lower bounds for the general case are also independent of n. Our goal is to understand whether the dependence on n is required in the multi-row case, making it fundamentally harder than the single-row version. We refute this by exhibiting an algorithm which is (1−e)-competitive as long as the right-hand sides are @math . Our techniques refine previous PAC-learning based approaches which interpret the online decisions as linear classifications of the columns based on sampled dual prices. The key ingredient of our improvement comes from a non-standard covering argument together with the realization that only when the columns of the LP belong to few 1-d subspaces we can obtain small such covers; bounding the size of the cover constructed also relies on the geometry of linear classifiers. General packing LP's are handled by perturbing the input columns, which can be seen as making the learning problem more robust.