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Virtual Network Embedding Approximations: Leveraging Randomized Rounding Virtual Network Embedding Approximations: Leveraging Randomized Rounding Matthias Rost Stefan Schmid TU Berlin, Germany University of Vienna, Austria Email: [email protected] Email: stefan [email protected] VM1 Abstract—The Virtual Network Embedding Problem (VNEP) LB1 Cache LB2 NAT captures the essence of many resource allocation problems VM2 VM5 of today’s infrastructure providers, which offer their physical Customer FW Internet VM VM computation and networking resources to customers. Customers 3 4 request resources in the form of Virtual Networks, i.e. as Fig. 1. Examples for virtual networks ‘in the wild’. The left graph shows a directed graph which specifies computational requirements a service chain for mobile operators [4]: load-balancers route (parts of at the nodes and communication requirements on the edges. the) traffic through a cache. Furthermore, a firewall and a network-address An embedding of a Virtual Network on the shared physical translation are used. The right graph depicts the Virtual Cluster abstraction for infrastructure is the joint mapping of (virtual) nodes to physical provisioning virtual machines (VMs) in data centers. The abstraction provides servers together with the mapping of (virtual) edges onto paths connectivity guarantees via a logical switch in the center [3]. in the physical network connecting the respective servers. This work initiates the study of approximation algorithms We study the offline setting with admission control: given for the VNEP. Concretely, we study the offline setting with admission control: given multiple request graphs the task is to multiple requests the task is to embed the most profitable embed the most profitable subset while not exceeding resource subset while not exceeding resource capacities. capacities. Our approximation is based on the randomized rounding of Linear Programming (LP) solutions. Interestingly, A. Formal Problem Statement we uncover that the standard LP formulation for the VNEP In the light of the recent interest in Service Chaining [2], exhibits an inherent structural deficit when considering general we extend the VNEP’s general definition [5] by consider- virtual network topologies: its solutions cannot be decomposed into valid embeddings. In turn, focusing on the class of cactus ing different types of computational nodes. We refer to the request graphs, we devise a novel LP formulation, whose solutions physical network as the substrate network. The substrate can be decomposed into convex combinations of valid embedding. GS = (VS;ES) is offering a set T of computational types. Proving performance guarantees of our rounding scheme, we This set of types may contain, e.g., ‘FW’ (firewall), ‘x86 obtain the first approximation algorithm for the VNEP in the server’, etc. For a type τ 2 T , the set V τ ⊆ V denotes the resource augmentation model. S S We propose two types of rounding heuristics and evaluate their substrate nodes that can host functionality of type τ. Denoting V τ performance in an extensive computational study. Our results the node resources by RS = f(τ; u) jτ 2 T ; u 2 VS g and all V indicate that randomized rounding can yield good solutions (even substrate resources by RS = RS [ ES, the capacity of nodes without augmentations). Specifically, heuristic rounding achieves and edges is denoted by dS(x; y) > 0 for (x; y) 2 RS. 73.8% of the baseline’s profit, while not exceeding capacities. For each request r 2 R, a directed graph Gr = (Vr;Er) I. INTRODUCTION together with a profit br > 0 is given. We refer to the Cloud applications usually consist of multiple distributed respective nodes as virtual or request nodes and similarly refer components (e.g., virtual machines, containers), which results to the respective edges as virtual or request edges. The types of virtual nodes are indicated by the function τ V !T . arXiv:1803.03622v2 [cs.NI] 4 May 2018 in substantial communication requirements. If the provider r : r Based on policies of the customer or the provider, the map- fails to ensure that these communication requirements are met, r;i τr (i) the performance can suffer dramatically [1]. Consequently, ping of virtual node i 2 Vr is restricted to a set VS ⊆ VS , while the mapping of virtual edge (i; j) is restricted to a over the last years, several proposals have been made to jointly r;i;j provision the computational functionality together with ap- subset of substrate edges ES ⊆ ES. Each virtual node propriate network resources. The Virtual Network Embedding i 2 Vr and each edge (i; j) 2 Er is attributed with a resource Problem (VNEP) captures the core of this problem: given a demand dr(i) ≥ 0 and dr(i; j) ≥ 0, respectively. Virtual nodes directed graph specifying computational requirements at the and edges can only be mapped on substrate nodes and edges r;i τr (i) of sufficient capacity, i.e. VS ⊆ fu 2 VS jdS(u) ≥ dr(i)g nodes and bandwidth requirements on the edges, an embedding r;i;j of this Virtual Network in the physical network has to be and ES ⊆ f(u; v) 2 ESjdS(u; v) ≥ dr(i; j)g holds. d r; x; y found, such that both the computational and the network We denote by max( ) the maximal demand that a request r may impose on a resource (x; y) 2 R : requirements are met. Figure 1 illustrates two incarnations of S r;i virtual networks: service chains [2] and virtual clusters [3]. dmax(r; τ; u) = max(f0g [ fdr(i)ji 2 Vr : τ(i) = τ ^ u 2 VS g) r;i;j A short version of this work appeared in IFIP Networking 2018 [6]. dmax(r; u; v) = max(f0g [ fdr(i; j)j(i; j) 2 Er :(u; v) 2 ES g) In the following the notions of valid mappings (respecting have been studied recently by considering restricted graph mapping constraints) and feasible embeddings (respecting classes for the virtual networks and the substrate graph. For resource constraints) are introduced to formalize the VNEP. example, virtual clusters with uniform demands are studied in [3], [12], line requests are studied in [13], [14], [15] and tree Definition 1 (Valid Mapping). A valid mapping m of re- r requests were studied in [14], [16]. quest r 2 R is a tuple (mV ; mE) of functions mV : V ! V r r r r S Considering approximation algorithms, Even et al. em- and mE : E !P(E ), such that the following holds: r r S ployed randomized rounding in [14] to obtain a constant ap- • Virtual nodes are mapped to allowed substrate nodes: V r;i proximation for embedding line requests on arbitrary substrate mr (i) 2 VS holds for all i 2 Vr. E graphs under strong assumptions on both the benefits and • The mapping m i; j of virtual edge i; j 2 E is r ( ) ( ) r the capacities. In their interesting work, Bansal et al. [16] an edge-path connecting mV (i) to mV (j) only using r r give an nO(d) time O(d2 log (nd))-approximation algorithm allowed edges, i.e. mE i; j ⊆ P Er;i;j holds. r ( ) ( S ) for minimizing the load of embedding d-depth trees based We denote by Mr the set of valid mappings of request r 2 R. on a n-node substrate. Their result is based on a strong LP Definition 2 (Allocations of Valid Mappings). We denote by relaxation inspired by the Sherali-Adams hierarchy. A(mr; x; y) the cumulative allocation induced by the valid To the best of our knowledge, no approximation algorithms mapping mr 2 Mr on resource (x; y) 2 RS: are known for arbitrary substrate graphs and classes of virtual X networks containing cyclic substructures. A(m ; τ; u) = d (i) 8(τ; u) 2 RV r V r S Bibliographic Note: This technical report is an extended i2Vr ,τ(i)=τ;mr (i)=u X version of our paper presented at IFIP Networking 2018 [6]. A m ; u; v d i; j 8 u; v 2 E ( r ) = E r( ) ( ) S (i;j)2Er ;(u;v)2mr (i;j) In our preliminary technical report [17] similar results were The maximal allocation that a valid mapping of request r 2 R presented. The current work presents a significantly simpler LP formulation and also provides an extensive computational may impose on a substrate resource (x; y) 2 RS is denoted evaluation. Additionally, in our recent technical report [18], by Amax(r; x; y) = maxm 2M A(mr; x; y). r r the approximation approach presented in this work is extended Definition 3 (Feasible Embedding). A feasible embedding beyond cactus request graphs. However, approximating more 0 of a subset of requests R ⊆ R is a collection of valid general request graphs comes at the price of non-polynomial mappings fmrgr2R0 , such that the cumulative allocations on runtimes. nodes and edges does not exceed the substrate capacities, i.e. P C. Outline of Randomized Rounding for the VNEP r2R0 A(mr; x; y) ≤ dS(x; y) holds for (x; y) 2 RS. Definition 4 (Virtual Network Embedding Problem). The We shortly revisit the concept of randomized rounding [19]. Given an Integer Program for a certain problem, randomized VNEP asks for a feasible embedding fmrgr2R0 of a subset of 0 P rounding works by (i) computing a solution to its Linear requests R ⊆ R maximizing the profit 0 br. r2R Program relaxation, (ii) decomposing this solution into convex B. Related Work combinations of elementary solutions, and (iii) probabilisti- In the last decade, the VNEP has attracted much attention cally selecting elementary solutions based on their weight. due to its many applications and the survey [5] from 2013 Accordingly, for applying randomized rounding for already lists more than 80 different algorithms for its many the VNEP, a convex combination of valid mappings k k k k variations [5].
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