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).
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.
@cite_5 @cite_10
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We develop a framework for proving approximation limits of polynomial size linear programs (LPs) from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any LP as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n1 2-ϵ)-approximations for CLIQUE require LPs of size 2nΩ(ϵ). This lower bound applies to LPs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by LPs. Our main technical ingredient is a quantitative improvement of Razborov’s [38] rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix. 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 the unsplittable flow problem (UFP) and the closely related column-restricted packing integer programs (CPIPs). In UFP we are given an edge-capacitated graph G = (V ,E ) and k request pairs R 1 , ..., R k , where each R i consists of a source-destination pair (s i ,t i ), a demand d i and a weight w i . The goal is to find a maximum weight subset of requests that can be routed unsplittably in G . Most previous work on UFP has focused on the no-bottleneck case in which the maximum demand of the requests is at most the smallest edge capacity. Inspired by the recent work of . [3] on UFP on a path without the above assumption, we consider UFP on paths as well as trees. We give a simple O (logn ) approximation for UFP on trees when all weights are identical; this yields an O (log2 n ) approximation for the weighted case. These are the first non-trivial approximations for UFP on trees. We develop an LP relaxation for UFP on paths that has an integrality gap of O (log2 n ); previously there was no relaxation with o (n ) gap. We also consider UFP in general graphs and CPIPs without the no-bottleneck assumption and obtain new and useful results. This paper addresses the problem of finding the minimum number of vehicles required to visit a set of nodes subject to time window and capacity constraints. The fleet is homogeneous and is located at a common depot. Each node requires the same type of service. An exact method is introduced based on branch and cut. In the computations, ever increasing lower bounds on the optimal solution are obtained by solving a series of relaxed problems that incorporate newly found valid inequalities. Feasible solutions or upper bounds are obtained with the help of greedy randomized adaptive search procedure (GRASP). A wide variety of cuts is introduced to tighten the linear programming (LP) relaxation of the original mixed-integer program. To find violated cuts, it is necessary to solve a separation problem. A substantial portion of the paper is aimed at describing the heuristics developed for this purpose. A new approach for obtaining feasible solutions from the LP relaxation is also discussed. Numerical results for standard 50- and 100-node benchmark problems are reported.