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IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT 1

A Path Generation Approach to Embedding ofVirtual Networks

Rashid Mijumbi, Joan Serrat, Juan-Luis Gorricho, Raouf Boutaba

Abstract—As the virtualization of networks continues to attractattention from both industry and academia, the Virtual NetworkEmbedding (VNE) problem remains a focus of researchers. Thispaper proposes a one-shot, unsplittable flow VNE solution basedon column generation. We start by formulating the problem as apath-based mathematical program called the primal, for whichwe derive the corresponding dual problem. We then propose aninitial solution which is used, first, by the dual problem and thenby the primal problem to obtain a final solution. Unlike mostapproaches, our focus is not only on embedding accuracy butalso on the scalability of the solution. In particular, the one-shotnature of our formulation ensures embedding accuracy, while theuse of column generation is aimed at enhancing the computationtime to make the approach more scalable. In order to assessthe performance of the proposed solution, we compare it againstfour state of the art approaches as well as the optimal link-basedformulation of the one-shot embedding problem. Experiments ona large mix of Virtual Network (VN) requests show that oursolution is near optimal (achieving about 95% of the acceptanceratio of the optimal solution), with a clear improvement overexisting approaches in terms of VN acceptance ratio and averageSubstrate Network (SN) resource utilization, and a considerableimprovement (92% for a SN of 50 nodes) in time complexitycompared to the optimal solution.

Index Terms—Network virtualization, resource allocation, vir-tual network embedding, column generation, optimization.

I. INTRODUCTION

The ever increasing requirements placed on the Internetare fueling its evolution to architectures which make a betterand more efficient use of the available network resources,and promote service innovations. Service Providers (SPs)have to satisfy personalized needs for their customers andhence they are impelled to use different protocol stacks andprovide customized services and network resources. Networkvirtualization [1] has been proposed as a feasible solutionto achieve this goal. In network virtualization, InfrastructureProviders (InPs) divide their resources into chunks, calledVNs, which are allocated to SPs. Thanks to virtualization, theresource chunks are isolated from each other so the servicenetworks behave as if they were independent though they sharethe same substrate infrastructure.

However, the creation of VNs on top of a SN is not trivial.VN topologies composed of virtual nodes and virtual linkshave to be drawn to support traffic flows from sources to sinks.Virtual nodes and virtual links then have to be mapped ontothe physical substrate in a way that satisfies user demands and

R. Mijumbi, J. Serrat and J.L. Gorricho are with the Network EngineeringDepartment, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain.

R. Boutaba is with the D.R. Cheriton School of Computer Science,University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada.

optimizes the use of the available resources. This is the basisof the so called VNE problem [1], which in case of unsplittableflows, i.e. flows that have to be treated as a unit from sourceto sink, is NP hard [2]. Therefore, to simplify the problem,several existing solutions to VNE either assume that the SNsupports the splitting of flows [3], or carry out the node andlink embedding in two separate steps [4], which can lead toblocking or rejecting of resource requests at the link mappingstage and hence a sub-optimal substrate resource utilization.

In this paper, we propose a near optimal solution to theunsplittable flow VNE problem obtained by performing theembedding in one-shot (i.e. both virtual nodes and linksare embedded in one step) using mathematical programmingand path generation1 [5]. The formulation of the embeddingproblem as being one-shot is motivated by the need to obtainan efficient embedding solution (which would ultimately leadto better resource utilization and hence better profitability forInPs), while the employment of path generation is aimedat ensuring that the resulting algorithm is more scalablecompared to the optimal formulation.

To this end, we formulate two mathematical programs;one is a path-based formulation of the unsplittable flowone-shot VNE problem, also known as the primal problem,while the other is its corresponding dual problem. For giveninstances of the problem, both the primal and dual problemshave approximately the same solution value. The proposedapproach begins by obtaining an initial solution (composedof paths in an augmented SN) to the primal problem usinga VNE approach that performs node and link mapping intwo coordinated stages. The next step is to enhance the initialsolution. This is achieved by using the initial solution as aninput into the dual problem, hence resulting into prices forthe SN links and nodes. Using Dijkstra’s algorithm [6], theseprices are utilized to determine an additional set of paths whichcan be added to enhance the solution. These paths, togetherwith those obtained in the initial solution, are finally used tosolve the primal problem to obtain a final embedding solution.

The main contributions of this paper are as follows:• A near optimal unsplittable flow one-shot VNE approach

that improves substrate resource utilization compared toexisting heuristic and approximation solutions.

• A path generation-based approach for unsplittable flowsthat significantly improves the time complexity of theembedding compared to the optimal solution.

The rest of the paper is organized as follows: Section II

1In this paper, we use the terms path generation and column generationinterchangeably.


presents the description of the VNE problem. We present therelated work in Section III. Sections IV and V respectivelydescribe the mathematical formulation of the one-shot embed-ding problem and its solution based on path generation. Sec-tion IV presents the evaluation of our proposed solution andthe discussion of the results. Finally, Section VII concludesthis paper.

II. PROBLEM FORMULATION

A. Substrate Network Capacity

We model the SN as an undirected graph denoted byGs = (Ns, Ls), where Ns and Ls represent the set of substratenodes and links, respectively. Each substrate link luv ∈ Ls

connecting the nodes u and v has a bandwidth capacity Cuv

while each substrate node u ∈ Ns has computation capacityCu and a location Locu(x, y)

B. Virtual Network Requests

In the same way, we model the VN as an undirected graphdenoted by Gv = (Nv, Lv), where Nv and Lv representthe set of virtual nodes and links respectively. Each virtuallink lij ∈ Lv connecting the nodes i and j has a bandwidthdemand Dij while each virtual node i ∈ Nv has computationdemand Di, a location Loci(x, y) as well a constraint on itslocation Devi(∆x,∆y) which specifies the maximum alloweddeviation for each of its x and y coordinates2. Constraintson the location of virtual nodes are aimed at giving SPs theflexibility to choose the geographical placement of given partsof their network topologies. This could be as a result of agiven SP introducing specialized services for users in a givenlocation, or a desire to ensure improved quality of service byrestricting the distance (and hence latency) between a givenpair of nodes.

C. Virtual Network Embedding

The embedding problem consists in the mapping of eachvirtual node i ∈ Nv to one of the possible substrate nodeswith in the set Υ(i). Υ(i) is defined as a set of all substratenodes u ∈ Ns which have enough available capacity (definedthe difference between the total capacity of a resource and theamount already allocated) and are located within the maximumallowed deviation Devi(∆x,∆y) of the virtual node. For asuccessful mapping, each virtual node must be mapped andany given substrate node can only map at most one virtualnode from the same request. Similarly, all the virtual linkshave to be mapped to one or more substrate links connectingthe nodes to which the virtual nodes at its ends have beenmapped. Each of the substrate links must have enough capacityto support the virtual link(s) that go through it. A mapping issuccessful if all the virtual links are mapped.

In Fig. 1, we show an example of two VNs being mappedonto a SN. The resource requirements for each virtual node orlink is also shown. The values in the SN are the total loadingof any given physical node or link. As can be noted from Fig.

2The notation used in this paper is to represent virtual nodes with the lettersi or j and substrate nodes with u or v.

Substrate Network

VN Request 1 VN Request 2

2

7

5

13

25

6

3

8

41

10

23

15

VNE

2

5+1 = 6 4

8

7+3 = 10

15

23

P

Q

A

B

R

Fig. 1: Virtual Network Embedding: Two VNs mapped onto a SN

1, one substrate node can host more than one virtual node (e.g.node A). A substrate link can also host more than one virtuallink (e.g. link AB), and a given virtual link can span morethan one substrate link (e.g link RP).

In general, the objective in VNE is to map as many VNsas possible, hence leading to an efficient utilization of SNresources. For the online VNE problem, there is no knowledgeon the requirements of future VN requests, and as such, oneway of ensuring that as many requests are accepted is bybalancing the overall loading of the SN [2] such that allsubstrate resources (nodes and links) are equally likely toaccept resource requests. In the same way, it is worth notingthat due to the lack of information about future virtual networkrequests, optimality as referred to in this paper is only basedon the mapping of an arriving virtual network request to asubstrate network, which could possibly already have othervirtual networks already embedded, or for which other virtualnetworks may be embedded in the future. Therefore, thisoptimality does not represent an optimal embedding solutionconsidering all possible virtual network requests.

III. RELATED WORK

VNE is a well-studied problem. In what follows, we onlydiscuss those approaches we consider more closely relatedto our proposal. An interested reader is referred to [1] fora detailed survey on VNE.

A. Two-step Embedding

Some approaches based on two stages, starting with nodemapping and then link mapping, are proposed in [3] and [7].These algorithms measure the resource of a node or linkby its CPU capacity, or bandwidth without considering thetopological structure of the VNs and the underlying substratenetwork. However, the topological attributes of nodes mayhave an impact on the success and efficiency of VNE. Chenget al. [8] propose a topology-aware node mapping approachwhich uses the Markov Random Walk model to rank virtualand substrate network nodes based on their resource andtopological attributes. The links are then mapped either usingthe shortest path (for unsplittable flows), or formulated asa commodity flow problem for splittable flows. Unlike our


work, the above approaches don’t consider location constraintson virtual nodes, assuming that they can be mapped at anylocation in the SN. A coordinated node and link mapping isproposed in [2]. Although the coordination here improves thesolution space, the mapping is still performed in two separatestages, hence yielding sub-optimal embedding.

The works in [9]–[12] propose dynamic and distributedapproaches to VN resource allocation, where the actual re-sources allocated to virtual nodes and links is scaled up anddown based on actual resources utilization as well as resourceavailability. However, they do not consider the embeddingstage, assuming that the VN is already mapped to a SN.

B. One-shot Embedding

A one-shot embedding solution based on a multi-agentsystem is proposed in [13]. However, this proposal assumesunbounded SN resources, and all VN requests to be knownin advance. Also, messaging overhead exchanged between theagents can be detrimental to solution scalability. Zhu et al. [4]also propose a one-shot mapping solution, assuming infinitesubstrate resources, and no constraints on the locations ofnodes. Authors in [14] - [15] propose different approachesto one-shot VNE assuming that all VN requests are known inadvance (offline solutions), while those in [16] - [17] makesimplifying assumptions with regard to the capacity of theSN and do not consider constraints on virtual nodes locations.While most VNE proposals use topologies to represent VNrequests, [18] proposes the use of traffic matrices. However,the embedding is achieved by alleviating constraints on VNresources, such as node location. Houidi et al. [19] splitany given VN request across multiple infrastructure providersand then uses max-flow and min-cut algorithms and linearprogramming to find one-shot solutions to the partial VNgraphs. While the embeddings of the split graphs are solvedin one-shot, they do not encompass the original VN requestin its entirety.

Perhaps the the works most related our work are by Jarrayet al. [20] and Hu et al. [21] both of whom apply columngeneration to VNE. Jarray et al. apply a column generationapproach to one-shot VNE by assuming that the embedding ofVN requests can be delayed by storing each arriving requestto process them in batches using an auctioning mechanism.The proposal can therefore be considered to be an offline one.Hu et al. formulate a one-shot path-based VNE where thevirtual links are represented as commodities. The formulatedmathematical program is then relaxed so as to apply columngeneration. However, Hu et al. consider a scenario where thedemand/commodity of any given virtual link may be split overmore than one substrate path. This differs from the proposalin this paper which solves a harder problem where the flowsare not splittable.

C. Mathematical Programming

Mathematical programming has been applied to a varietyof problems in networking. Xie et al. [22] use mathematicalprogramming for dynamic resource allocation in networkswhile Botero et al. [23] use an optimization technique for

link mapping (assuming that the virtual nodes have alreadybeen mapped to substrate nodes). Unlike all these works,the mathematical programming formulation proposed in thispaper does not only focus on unsplittable flows, but alsocombines both node and link mapping in one stage. Thenode mapping step is an important part of VNE since itdetermines the efficiency of the link mapping. This is whysuch approaches that coordinate these two steps have beenshown to lead to better resource utilization efficiency [2].Combining these two steps together even further enhancesthis efficiency, yet the resulting mathematical program is evenharder to solve. Finally, path generation based formulationsfor multi-commodity flow based problems are proposed in anumber of approaches such as [24]. In these formulations thesource and end nodes for each flow are known a priori, whichreduces the complexity of the problem, compared to the one-shot VNE that we solve in this paper.

D. SummaryTo summarize, because of the NP hardness of the VNE

problem, existing one-shot approaches either make simplifyingassumptions such as considering the offline version of theproblem, assuming infinite resources, or ignoring constraintson the virtual nodes and links, while other proposals solvethe embedding problem in two stages, typically employing agreedy approach for node mapping and then try to optimizethe link mapping. The approach proposed in this paper differsfrom previous work in many aspects. Most applications ofmathematical programming and path generation to routing areconcerned with simpler problems, in which either both thesource and sink nodes are known, in which case the problemreduces to a load balancing problem, or only consider nodemapping. While the link mapping sub-problem is still NP-hard for unsplittable flows, it is even harder in the caseof VNE since the source and sink nodes should also bedetermined. To the best of our knowledge, this is the firstpath-based mathematical programming solution to the one-shot, unsplittable flow VNE problem.

IV. ONE-SHOT VIRTUAL NETWORK EMBEDDING

The one-shot VNE problem involves performing both nodeand link mapping at the same time. In this paper, we usemathematical programming to achieve this. Specifically, weconsider that VNs arrive one at a time following a Poissondistribution and have exponentially distributed service times,and the formulated optimization problem involves the embed-ding of a single VN at any given time. This way, at everymapping step, the actual resource availability of all substratelinks and nodes is taken into account when performing amapping. For reference, the link-based formulation of theproblem that obtains an optimal solution using mathematicalprogramming is shown in the appendix. Here, we adopt a path-based formulation using column generation in order to solvethe problem with much less time and storage requirements.

A. Substrate Network AugmentationWe start by creating an augmented network first introduced

in [2], with each virtual node i connected to each of the


C

B

G

X

E

F

Y

D

A

Z

Substrate Links

Virtual Links

Substrate Nodes

Virtual Nodes

Possible virtual to substrate node mappings

Fig. 2: VNE showing Virtual Node Mapping Constraints

substrate nodes in its possible node set Υ(i) by a meta link[2] liu ∈ Lx, where Lx is the set of all meta links. Then theaim is to establish a single path pijuv from each virtual node ito all other virtual nodes j to which it is connected. The pathpijuv is made of two meta links, liu and ljv , and a sub-path inthe SN connecting the substrate nodes u and v. This sub-pathmay be made up of one or more SN links. In Fig. 2, we showa representation of an instance of the problem. In the figure,XYZ are nodes of a VN, while ABCDEFG are nodes of aSN. As an example, for virtual link XZ, one possible pathcould be XABEZ, and is represented as pxzae . The path pxzae isa sequence of links in the augmented network that start fromone end of the virtual link to the other. Therefore, in orderto embed the virtual link XZ, we need to determine the threecomponents of the path, which − for this example − are thetwo meta links XA and EZ, and the SN path ABE composedof two links, AB and BE. The components XA and EZ can bedetermined from a virtual to substrate node mapping, whileABE from a link mapping approach such as shortest path.In particular, this path example would mean that the virtualnode X is mapped onto substrate node A, the virtual node Zis mapped onto substrate node E and that the virtual link XZis mapped onto the SN path ABE. One difficulty illustratedin this example comes from the fact that if, for example, wechoose the path XABEZ for virtual link XZ, then the virtuallink XY can only be mapped on a path that includes meta linkXA and not XC. This would in turn require that Y be mappedonto C, otherwise we would have a sub-optimal solution inwhich the virtual link XY uses resources from two substratelinks (AC & CG) instead of a single link (CG). Hence, thedetermination of these paths should not be carried sequentiallyand independently. As previously mentioned, our aim is to findthe best possible path for each of the virtual links subject to themapping requirements described in our problem formulation(see Section II).

B. LP−P: Path based Formulation −PrimalWe formulate the VNE problem as a commodity flow

problem [25], where virtual links are flows that should becarried by the SN. However, unlike most commodity flowformulations, in our case, the source node i and terminal

node j for each flow also need to be determined.

Variable and Parameter definitions: In this formulation, wedefine a non-negative binary variable f ijuv = [0, Dij ] whichrepresents the unsplittable flow of a virtual link lij ∈ Lv

on a simple substrate path pijuv ∈ P . The indices u, v, iand j define a path (i − u − v − j) in the augmented SN.As described in IV (A), these paths are made up of threecomponents: two meta-links iu and jv, and a SN path fromu to v. The variable f ijuv is binary in that it can only take onvalues 0 and Dij , where Dij is the demand of virtual linklij ∈ Lv . We define P as a set of all the possible substratepaths, Puv as the set of all paths that use the substrate linkluv ∈ Ls and P ij as the set of all paths that can support theflow for virtual link lij ∈ Lv . We also define χi

u = [0,1] as abinary variable equal to 1 if the virtual node i is mapped ontothe substrate node u and 0 otherwise. As mentioned in IV(A), it is important to note that variables χi

u and χjv directly

determine the existence or otherwise of meta links iu and jvfor the path pijuv since the meta links are dependent on therespective node mappings. For example, if χi

u == 0 then thevirtual node i is not mapped onto substrate node u, implyingthat the meta link from i to u is non existent, and so is thepath pijuv . Let Auv be the available bandwidth capacity onthe substrate link luv , and Au be the available computationcapacity on node u.

Objective: The objective of the mathematical formulation(1)−(7) is to balance the resource usage of the SN, byfavoring the selection of those resources with comparativelyhigher available capacity. Balancing the loading of the SNhas two advantages; first, it distributes the mapping of a givenVN request over multiple SN resources which avoids a singleVN being majorly affected by single or regional failures inSN, hence ensuring better VN survivability. In addition, sincethe problem we consider in this paper is online, we do notknow in advance the required node locations for VN requests.Balancing the loading of the SN ensures that at any givenpoint, each substrate node/link has the same capacity onaverage. This avoids situations where a VN request would berejected due to one or more of its nodes not being able to bemapped because substrate nodes in their respective possiblenode sets Υ(i) have less resources than other parts of theSN. As was shown by [2], load balancing leads to a betteracceptance ratio of VNs, which would directly translate inhigher incomes for InPs.

minimize∑

lij∈Lv

∑pijuv∈P

1

Auvf ijuv +

∑i∈Nv

∑u∈Υ(i)

1

Auχiu (1)

subject to ∑u∈Υ(i)

χiu = 1 ∀i ∈ Nv (2)

∑i∈Nv

χiu ≤ 1 ∀u ∈ Ns (3)

∑pijuv∈P ij

f ijuv = Dij ∀lij ∈ Lv (4)


∑pijuv∈Puv

f ijuv ≤ Auv ∀luv ∈ Ls (5)

f ijuv −Dijχiu ≤ 0 ∀pijuv ∈ P (6)

f ijuv −Dijχjv ≤ 0 ∀pijuv ∈ P (7)

f ijuv = [0, Dij ] ∀pijuv ∈ P

χiu = [0, 1] ∀i ∈ Nv,∀u ∈ Ns

The first term in the objective (1) is for link mapping, whilethe second term is for node mapping. Each of these terms aredivided by the respective capacities to ensure that the substrateresources with more free resources are preferred. Constraint(2) ensures that each virtual node is mapped to a substratenode, while (3) ensures that any substrate node may be used atmost once for a given mapping request. Constraints (4) and (5)represent the virtual link demand requirements and substratelink capacity constraints respectively. Specifically, (4) statesthat the flow f ijuv on path pijuv should carry the total demand ofthe virtual link ij, while (5) states that the flow f ijuv on path pijuvshould be at most equal to the capacity of each substrate linkon that path. From constraint (6), if χi

u == 0 then f ijuv = 0.If χi

u == 1 then f ijuv = [0, Dij ]. This is also true for (7).These constraints ensure that virtual links and virtual nodesare mapped at the same time, i.e., a flow f ijuv − using the pathpijuv starting with meta link iu and ending with meta link jv −is only non-zero if the virtual node i is mapped onto substratenode u and j is mapped onto v. Together, (6) and (7) ensurethat a flow f ijuv is only non zero if both the two end links iuAND jv exist.

The formulation in (1)−(7) is intractable for two reasons;first, the restrictions that variables χi

u and f ijuv only take onbinary values, and then the fact that the number of possiblepaths pijuv (and hence the number of variables f ijuv) is very large(exponential) even for moderately sized networks. Therefore,solving the problem in its current form is impractical. Thereare three possibilities to solving the problem;

1) a relaxation to the constraints on variables χiu and f ijuv

to take on continuous values,2) restricting the number of input variables f ijuv (by restrict-

ing the number of paths pijuv).3) a combination of both the first two approaches.

For the VNE problem as formulated in (1)−(7), a relaxationwould require careful consideration to avoid violating therequirements that both nodes and links are mapped in one-shot (since the variables χi

u would no longer be able to restrictthe mapping of virtual nodes to particular substrate nodes), aswell splitting the flows of the virtual links across multiplelinks. Therefore, we take the second approach, and employpath generation, which allows for the use of only a sufficientlymeaningful number of paths, and adding more paths as neededuntil a final solution is obtained.

V. PATH GENERATION

Path generation is a method that solves mathematicalprograms with a large number of variables efficiently. Themain idea is to solve a restricted version of the program(the restricted primal problem [26]) - which contains only asubset of the variables, and then (through the use of the dualproblem [26]) add more variables as needed. Usually, pathgeneration involves creating an initial solution (restricted setof variables) which are used in the solution for the restrictedprimal problem. Then, solving pricing problems (which aredetermined from the dual problem), allows for adding morevariables to improve the initial solution, until either a finaloptimal solution is found, or a stopping condition is reached.

The path generation approach taken in this paper is asfollows: we start by creating an initial set of paths using atwo stage node and link mapping. We then use these pathsto solve a dual problem, and use the pricing problems todetermine a set of paths to add to the initial solution. Thesepaths are then used to solve a restricted primal problem toobtain a final solution. Therefore, our proposal avoids theusual iteration required in a path generation approach wherethe primal and dual problems are solved sequentially, manytimes, instead preferring only to perform a single iteration. Inthe next subsections, we propose a method for determiningthe initial set of paths, derive the pricing problems, and thendescribe the overall algorithm proposed in this paper.

A. Initial Solution

An initial solution (Init−Sol) is determined as a set ofpaths P ′ in the augmented SN, with each path pijuv ∈ P ′

able to support the flow f ijuv of virtual link lij . Each of thesepaths must be able to meet the VN mapping conditions asformulated in the primal problem. Considering the examplein Fig. 2, since we have two virtual links, an initial solutionwould have two paths, one for each virtual link. Examples ofthese paths could be XABEZ and XACY for virtual links XZand XY respectively. In order to determine such a path, sayfor virtual link XZ, the approach in this paper is as follows:we start by performing a node mapping, which for thisexample, would map virtual nodes X and Z onto substratenodes A and E respectively. This step gives us the meta linksXA and EZ. In this subsection, we propose a novel nodemapping solution LP−N for determining XA and EZ. Thenext step involves determining the path ABE in the SN. Thisis done by using Dijkstra’s algorithm, with the constraintthat each link on the path should have enough capacity tosupport the virtual link under consideration. The completepath is determined by joining meta links XA and EZ to therespective ends of ABE.

LP−N: Node Mapping: LP−N is based on mathematicalprogramming. It is formulated in such a way that mappingof any given virtual node is relatively biased towards eachsubstrate node by a weight. The determination and use of theweights is discussed in what follows.

Objective: It is noteworthy that, essentially, LP−N is


10

20 70 50

50 60

G

Y

A

C

X

Fig. 3: Node-Link Weighted Averages

aimed at achieving an initial node and link mapping. Assuch, many other state-of-art two-step approaches [1] couldbe used for this purpose. However, the authors could not finda previous two-stage mapping approach that simultaneouslyachieves both objectives considered in our formulation: Thefirst objective is to keep the computation time of the initialsolution as low as possible by including only the possiblevirtual node to substrate node combinations. Secondly, asexplained later in this section, we minimize the possibilityof failure at the link mapping stage, by making the nodemapping aware of the link mapping stage through the use ofweights Wi and Wu.

Variable definition: As before, χiu is a binary variable

equal to 1 when the virtual node i is mapped onto substratenode u and 0 otherwise.

minimize∑i∈Nv

∑u∈Υ(i)

Wi

Wuχiu (8)

subject to: ∑u∈Υ(i)

χiu = 1 ∀i ∈ Nv (9)

∑i∈Nv

χiu ≤ 1 ∀u ∈ Ns (10)

χiu = [0, 1] ∀i ∈ Nv,∀u ∈ Ns

Constraints (9) and (10) are the same as (2) and (3). Theweights Wi and Wu are dynamically determined for eachvirtual and substrate node respectively. The motivation to usesuch weights is from the need bias or coordinate the mappingof the nodes to the following link mapping step. This has beenshown by related works to improve the mapping efficiency [2]by avoiding the use of a high amount of resources for the linkmapping phase. In our proposal, this is particularly importantto avoid the possibility that we fail to obtain an initial solutiondue to unavailable SN resources. Therefore, Wu is definedas the weighted average of the available capacities of all thesubstrate links connected to u. Similarly, Wi is defined asthe weighted average of the demand of all the virtual linksconnected to i. To illustrate the idea behind these weightedaverages, consider Fig. 3, which is a subset of the topologyrepresented in Fig. 2. The values beside each link represent

the available link bandwidths and link demands respectively.As an example, considering the virtual node X,

WX = 20×

(20

20 + 10

)+ 10×

(10

20 + 10

)= 16.67.

In the same way for substrate node C,

WC = 50×

(50

50 + 60 + 70

)+ 60×

(60

50 + 60 + 70

)+

70×

(70

50 + 60 + 70

)= 61.11

The reason for using this ratio as a weight is to ensure thatthose substrate nodes that are connected to many substratelinks with higher available resources are usually preferred, andthat in case two or more virtual nodes have a given substratenode in their possible node set (such as X and Y in Fig. 2), thenthe substrate node would always be allocated to that virtualnode with the highest weighted average link demand. Thisachieves some level of coordination between the node mappingand link mapping phases and thereby reduces the probabilityof rejecting link mapping requests.

We note that there could be instances where the weightedaverages lead to selecting substrate nodes with less good links,especially when the links have widely differing residual capac-ities. For example, a node connected to two links with residualcapacities 80 and 10 respectively will have a W1 = 72, whilea node connected to two links with residual capacities 60 and70 respectively will have a W2 = 65. In this case, the firstnode will be selected yet the second node could be a betterchoice. One simple solution to handle such scenario is to usethe sum of two averages: the weighted average and a simpleaverage. However, it is worth mentioning that in our approachnetwork embedding is done in such a way that the averageloads of SN nodes and links are balanced, this way, avoidingscenarios where some node and/or links have widely differingresidual capacity. The procedure, Init-Sol, for determining theinitial solution is shown in Algorithm 1.

B. Pricing Problem

To determine which paths should be added to the initial setso as to improve the solution, we need to solve the pricingproblems for LP−P. In order to identify the pricing problemswe first formulate the dual problem LP−D for the primalproblem LP−P. The formulation of a dual problem from aprimal can be obtained in five steps [27].

1) Creating a dual variable for every primal constraint,2) Creating a dual constraint for every primal variable,3) The right-hand sides of primal constraints become coef-

ficients for the dual objective,4) The coefficients of the primal become right-hand sides

of the dual constraints,5) If the primal problem is a maximisation problem, the

dual is a minimisation problem.In Table I, we summarize these steps, giving the bounds forthe resulting dual constraints and variables for all possiblecases of primal variables and constraints respectively. These


Algorithm 1 Init−Sol (Gv(Nv, Lv), Gs(Ns, Ls))

1: for i ∈ Nv do2: Determine Candidate Node Set, Υ(i)3: if Υ(i) = ∅ then4: Reject Request5: end6: end if7: Calculate Wi

8: end for9: for u ∈ Ns do

10: Calculate Wu

11: end for12: Solve : LP-N13: for lij ∈ Lv do14: for u ∈ Υ(i) do15: if χi

u = 1 then16: Meta Link 1: l1 = iu17: Start Node, s = u18: end if19: end for20: for v ∈ Υ(j) do21: if χj

v = 1 then22: Meta Link 2: l2 = jv23: End Node, t = v24: end if25: end for26: LinkMapping: ps = Dijkstra

(s, t,Gs(Ns, Ls)

)27: Create Path: pijuv = l1 + ps + l228: Add pijuv to P ′

29: end for

conventions reflect the interpretation of the dual variablesas shadow prices of the primal problem. A less-than-or-equal-to constraint, normally representing a scarce resource,has a positive shadow price, since the expansion of thatresource generates additional profits. On the other hand,a greater-than-or-equal-to constraint usually represents anexternal requirement (e.g., demand for a given resource).If that requirement increases, the problem becomes moreconstrained; this produces a decrease in the objective functionand thus the corresponding constraint has a negative shadowprice. Finally, changes in the right hand side of an equalityconstraint might produce either negative or positive changesin the value of the objective function. This explains theunrestricted nature of the corresponding dual variable.

Dual Variables definitions: To determine the dual program,we start by relaxing the bounds of the variables χi

u and f ijuvsuch that χi

u ≥ 0 and f ijuv ≥ 0. Then, following the fivesteps stated above, we define six dual variables as follows: λifor the virtual node constraints (2), µij for the virtual linksdemand constraints in (4), ηu >= 0 substrate node constraintsin (3), γuv >= 0 substrate links available capacity constraintsin (5), σiu >= 0 for simultaneous node and link mappingconstraint constraint (6) and τjv >= 0 for constraint (7).Since most results of duality for linear programs do extend to

TABLE I: Relationship between dual and primal problems

Primal Dual

Objective Function

Maximisation Minimisation

Variable bounds Constraint bounds

−∞ ≤ i ≥ +∞ i =

i ≥ 0 i ≥i ≤ 0 i ≤

Constraint bounds Variable bounds

j = −∞ ≤ j ≥ +∞j ≥ j ≤ 0

j ≤ j ≥ 0

We use A←X to mean that node virtual node X is mapped

onto substrate node A

(A←X, B←Z), (A←X, E←Z), (A←X, D←Z)

(C←X, B←Z), (C←X, E←Z), (C←X, D←Z)

B

D Z

X

E

C

A

Fig. 4: Possible substrate node combinations for virtual link XZ

integer programming [28], the dual formulation in this paperis based on [27].

The objective of the dual formulation (11)−(13) is to obtaina mathematical program that produces a maximized value asclose as possible to that of its original primal program for anyinstance of the variables. Therefore, the dual of the primalformulation in (1)−(7) is:

maximize∑i∈Nv

λi +∑

lij∈Lv

Dijµij −∑u∈Ns

ηu−∑

luv∈Ls

Auvγuv

(11)subject to

λi +∑

pijuv∈P

(σiu + τjv)− ηu ≤1

Au∀liu ∈ Lx (12)

µij−σiu−∑

luv∈pijuv

γuv−τjv ≤∑

luv,liu∈pijuv

1

Auv∀pijuv ∈ P

(13)The pricing problems are shown in (12) and (13). From (12),

the pricing condition for substrate nodes can be determined as:

λi +∑

pijuv∈P

(σiu + τjv) >1

Au+ ηu

However, since the variables χiu are much fewer compared

to f ijuv , we include all the possible substrate nodes for each


virtual node in the restricted primal problem. This eliminatesthe need for node pricing and we are left to deal with onlythe link pricing problem (13):

µij >∑

luv,liu∈pijuv

1

Auv+ (σiu +

∑luv∈pij

uv

γuv + τjv)

This pricing problem can be solved using the shortest pathalgorithm. Any path pijuv = Siu + (luv ∈ Puv) + Tjv in theaugmented SN whose length with respect to the dual variables(this means that the costs of the substrate links luv ∈ Puv areγuv , those of meta links Siu are σiu and those of Tjv areτjv) is smaller than µij satisfies the inequality above, and hasthe potential to improve the solution. However, a change inpath for any given virtual link could necessitate a change inthe mapping of one of its end nodes, which would changethe prices and feasibility of mappings for other virtual linksconnected to it. For example in Fig. 2, if the virtual node X ismapped onto substrate node C, all the paths for both links XZand XY go through C. If the path for say XZ is changed to gothrough A, it would either mean that the path for XY shouldalso be changed to go through A, otherwise this path cannotbe used to give a feasible and improved solution. Therefore,addition of paths individually for each virtual link does notguarantee that each of the added paths would still lead toa feasible solution, and for as long as the added path cannotyield a feasible solution, this path cannot lead to improvementin the solution of the restricted primal problem. In this case,there would be no guarantee that the pricing problems can besolved in polynomial time, as it could require quite a numberof iterations before enough paths are added to actually improvethe solution.

In this paper, instead of adding individual paths for eachvirtual link in each iteration of the path generation algorithm,we include all the possible shortest path combinations aftersolving the formulation (11) − (13). We use Fig. 4, whichis extracted from Fig. 2, to illustrate this for the case ofvirtual link XZ. Since the node X has two possible substratenodes and virtual node Z has three possible substrate nodes,then the possible combinations for these nodes are 6. In ourpricing solution, we determine the shortest path − based onthe weights in (14) for each of these 6 possible end nodecombinations.∑

luv,liu∈pijuv

1

Auv+

(σiu +

∑luv∈pij

uv

γuv + τjv

)(14)

This is done for all the virtual links, and all the correspondingpaths are added to the restricted primal problem. However,the number of paths added for each pricing iteration would betoo big to handle if many iterations are carried out. Even theDijkstra algorithm takes quite some time to find the shortestpaths. For this reason, we perform only one round for thesubstrate paths and use the resulting shortest paths based onthe dual problem to solve LP−P to obtain the final solution.As we show in the simulation results, the solution obtained isnear optimal.

The proposed approach, Final−Sol, for determining thefinal solution is shown in Algorithm 2.

Algorithm 2 Final−Sol(Gv(Nv, Lv), Gs(Ns, Ls))

1: Create Augmented Substrate Network2: Initial Paths Set: P ′ ← Solve Init−Sol3: Solve LP−D(P ′)4: for lij ∈ Lv do5: for u ∈ Υ(i) do6: for v ∈ Υ(j) do7: P ′ ← (P ′ +GetShortestPath(i, u, v, j))8: end for9: end for

10: end for11: Solve LP−P(P ′)

Example: To illustrate the details of Final−Sol in algorithm2, we use a simple running example based on Fig. 2 as wellas the flow diagram in Fig. 5a. The aim of the example isto illustrate the sequence of the proposed algorithm ratherthan its effectiveness, which is evaluated in the next section.As such, we keep it simple by avoiding the details of howthe actual mathematical programs are solved. In Fig. 5b, weshow a possible initial solution (black dotted lines) wherevirtual nodes X, Y and Z have been mapped to substratenodes A, G and E respectively. The virtual links XY andXZ have been mapped onto substrate paths ACG and ABErespectively. Therefore, based on the discussion in sectionIV(A), the initial solution is made up of two paths XACGYand XABEZ in the augmented substrate network. In Fig.5a, these two paths make up P1. With these paths, the dualproblem (LP-D(P1)) is solved. The values of σiu, γuv andτjv along each link in Fig. 5b represent hypothetical valuesthat could result from solving LP-D. As explained above, andillustrated in Fig. 4, the next step is then, for each virtuallink, to find the shortest path in the substrate network for allthe possible virtual-to-substrate node mappings. Using thevalues in Fig. 5b, the these shortest paths are determinedusing Dijkstra’s algorithm as shown in Table II3 for the 6combinations (see Fig. 4) of the virtual link XZ. Using asimilar process, the paths corresponding to the virtual linkXY are determine. These paths (excluding those which werealready in the initial solution such as XABEZ) constitute P2

in Fig. 5a. The combined paths (P1 + P2) are then used asinputs to solve a restricted primal problem to obtain a finalsolution.

VI. PERFORMANCE EVALUATION

A. Simulation Setup

To evaluate the performance of our proposed approach, weimplemented a discrete event simulator in Java, which usesthe tool Brite [29] to generate substrate and VN topologies.We used the tool ILOG CPLEX 12.4 [30] to solve themathematical programs. Simulations were run on Windows 8

3The reader should note that for representation simplicity, the terms 1/Auv

in (14) are not included in the shortest path summations in Table II. Theseterms represent the reciprocal of the available bandwidth on each link alongthe shortest path in the augmented substrate network.


Final VNE Solution

DjikstraShortest Paths

VN Request

Initial VNE Solution

Paths, P1

Prices

Paths, P2Paths, (P1+P2)

Paths, P1

Dual Problem

Solution

Primal Problem Solution

1 2

34

(a) Summary of Path generation-based VNE Approach

B

X

E A

Z

C D

Y

G F

𝜇𝑋𝐴 = 1

𝜇𝑋𝐶 = 3

𝜇𝑌𝐶 = 1

𝜇𝑌𝐺 = 2

𝜏𝐵𝑍 = 1.5

𝜏𝐸𝑍 = 3

𝜏𝐷𝑍 = 2

𝛾𝐶𝐺 = 4

𝛾𝐴𝐶 = 3

𝛾𝐴𝐵 = 2

𝛾𝐵𝐸 = 5

𝛾𝐷𝐹 = 5

𝛾𝐶𝐷 = 4

𝛾𝐵𝐷 = 2

(b) Initial Solution and Dual Pricing of Links

Fig. 5: Running Example

TABLE II: Shortest Paths for Virtual Link XZ

Path Values Along Path Total Path Length

XABEZ 1 + 2 + 5 + 3 11

XABZ 1 + 2 + 1.5 4.5

XABDZ 1 + 2 + 2 + 2 7

XCZBEZ 3 + 3 + 2 + 5 + 3 16

XCABZ 3 + 3 + 2 + 1.5 9.5

XCDZ 3 + 4 + 2 9

TABLE III: Brite Network Topology Generation Parameters

Parameter Substrate Network Virtual Network

Name (Model) Router Waxman Router Waxman

Number of nodes (N) 100 and 20 [15-25] and [3-10]

Size of main plane (HS) 500 500

Size of inner plane (LS) 500 500

Node Placement Random Random

GrowthType Incremental Incremental

Neighbouring Nodes 3 2

alpha (Waxman Parameter) 0.15 0.15

beta (Waxman Parameter) 0.2 0.2

BWDist Uniform Uniform

TABLE IV: Performance Quality Evaluation Algorithms

Code Mapping Method

GNMSP Greedy Node Mapping and Shortest Path (SP) for Links [3]

CNMMCF Coordinated Node and MCF for Link Mapping [2]

VNA-1 One-Shot Mapping [4]

TANMSP Topology-aware Node Mapping and SP for Links [8]

PaGeViNE Path Generation based Virtual Network Embedding

ViNEOPT Link based Optimal Virtual Network Embedding

Pro running on a 4.00GB RAM, 3.00GHz Processor Machine.Both substrate and VNs were generated on a 500× 500 grid.The CPU and bandwidth capacities of substrate nodes andlinks are uniformly distributed between 50 and 100 unitsrespectively. The CPU demand for VN nodes is uniformly dis-tributed between 2 and 10 units while the bandwidth demand

of the links is uniformly distributed between 10 and 20 units.The parameters used in Brite to generate network topologiesare shown in Table III. The parameters α and β are Waxman-specific exponents, such that, 0 < α ≤ 1, 0 < β ≤ 1, (α, β) ∈R. α represents the maximal link probability while β is usedto control the length of the edges. High values of alpha leadto graphs with higher edge densities while high values of betalead to a higher ratio of long edges to short ones. The valuesused in this paper are the default values in the Brite routerWaxman model used in [29]. Each virtual node is allowed tobe located within a uniformly distributed distance between 100and 150 units of its requested location. For embedding qualityevaluations, two possible sets of network sizes have been used.One involves a SN with 100 nodes and VNs with number ofnodes varied uniformly between 15 and 25, while the otherhas a SN with 20 nodes and VNs with number of nodesvaried uniformly between 3 and 10. The need for differentnetwork sizes will be explained in a later subsection. Forthese simulations, we assumed Poisson arrivals at an averagerate of 1 per 3 time units. The average service time of therequests is 60 time units and assumed to follow a negativeexponential distribution. The experiments are performed for1500 arrivals. For the time complexity evaluation, the numberof nodes for the SN is gradually increased from 20 to 100,and each simulation setup is repeated 20 times and averagevalues determined.

B. Performance Metrics

1) Solution Quality: Three performance indicators − Ac-ceptance ratio, Node utilization and Link utilization − are usedfor quality evaluation. The acceptance ratio gives a measureof the number of VN requests accepted compared to the totalrequests. We define the average node utilization as the averageproportion of the total substrate node capacity that is under useat any given time. In the same way, we define average linkutilization as the average proportion of the total substrate linkcapacity that is under use at any given time.

2) Solution Complexity: We define the time complexityof a given solution as the average time to complete thecomputation.

3) Embedding Cost and Revenue: We define the costs andrevenue from embedding a given VN the same way as a related


0.0

0.2

0.4

0.6

0.8

1.0

100 400 700 1000 1300

Acce

pta

nce

Ra

tio

Number of Arrivals

GNMSP VNA-I TANMSP

CNMMCF PaGeViNE ViNEOPT

Fig. 6: Average Acceptance Ratio - 20 SN Nodes

Acc

epta

nce

Rat

io

0

0.2

0.4

0.6

0.8

1

100 400 700 1000 1300

Accepta

nce R

atio

Number of Arrivals

GNMSP VNA-I TANMSP

CNMMCF PaGeViNE

Fig. 7: Average Acceptance Ratio - 100 SN Nodes

0.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5

Acce

pta

nce

Ra

tio

Arrival Rate

GNMSP VNA-I

TANMSP CNMMCF

PaGeViNE ViNEOPT

Fig. 8: Effect of VN Arrival Rate on Acceptance Ratio

0.0

0.1

0.2

0.3

0.4

0.5

100 400 700 1000 1300

No

de U

tilis

atio

n

Number of Arrivals

GNMSP VNA-I TANMSP


Fig. 9: Average Node Utilisation

0.0

0.2

0.4

0.6

100 400 700 1000 1300

Lin

k U

tilis

atio

n

Number of Arrivals

GNMSP VNA-I TANMSP


Fig. 10: Average Link Utilisation

0.0

0.2

0.4

0.6

1 2 3 4 5

Lin

k U

tilis

atio

n

Arrival Rate

GNMSP VNA-I TANMSP


Fig. 11: Effect of VN Arrival Rate on Link Utilization

work [2]. In particular, we define revenue, R(Gv(Nv, Lv)

)as the benefit to the SN for accepting the VN requestGv(Nv, Lv). As formulated in (15), it is the weighted sumof the link and node demands for the VN.

R

(Gv(Nv, Lv)

)=∑i∈Nv

Di +∑

luv∈Lv

Dij (15)

Similarly, in (16), we define an embedding cost

C

(Gv(Nv, Lv)

)as the sum of total substrate resources

that are allocated to the VN Gv(Nv, Lv). κu and ξuv areparameters that represent the relative unit costs of substratenodes and links respectively, where the virtual nodes and linksare mapped.

C

(Gv(Nv, Lv)

)=∑i∈Nv

κuDi +∑

lij∈Lv

∑luv∈Ls

ξuvfijuv (16)

C. Comparisons

We compare the performance of our solution with closelyrelated solutions. In particular, four representative solutionsfrom the literature are chosen. We name and describe thecompared solutions in table IV. These solutions were slightlymodified to fit into our formulation of the problem. Specif-ically, unsplittable flows, constraints on SN capacities andconstraints on virtual node locations were applied. We alsoimplemented a baseline formulation of the optimal one-shotmapping (see Appendix).

Since ViNEOPT requires a very long time (in excess of1 hour for a single embedding involving a SN of 60 nodesand a VN of 10 nodes) to perform an embedding, simulationsevaluating this algorithm have been restricted to SNs with 20

nodes and VNs with nodes from 3 − 10. However, an extrasimulation for acceptance ratio using larger sized networks hasbeen performed so as to reflect more practical network sizes.This simulation excludes ViNEOPT.

D. Results

1) Solution Quality: From the graphs in Fig. 6 it is evidentthat PaGeViNE achieves an average acceptance ratio close tothat obtained by the optimal solution ViNEOPT. In additionFig. 6 and Fig. 7 show that PaGeViNE outperforms state-of-the-art solutions in terms of average acceptance ratio. Thesetwo figures also confirm that the embedding efficiency ofPaGeViNE is not affected by increasing the size of substrateand VNs. In addition, it can be observed from Fig. 8 thateven as the arrival rate of VNs increases, PaGeViNE continuesto perform comparable to ViNEOPT and better than thefour other approaches. The fact that CNMMCF is under-performing PaGeViNE with respect to the average acceptanceratio and resource utilization can be attributed to the factthat CNMMCF is using more resources at the link mappingstage since it performs node and link mappings separately.For VNA-1, while the node and link mapping is done in oneshot, they are carried out sequentially, considering specificclusters of the SN each time. It is therefore expected that theresults would not be as good as those achieved by a globalsolution based on mathematical programming. It can also beobserved that TANMSP, which uses the topology informationto determine node mapping performs better than VNA-1 andGNMSP. However, since it still falls short of CNMMCF whichdetermines the node mapping from a mathematical program.

It is also evident from the graphs in figures 9 and 10that PaGeViNE achieves a better utilization ratio for substratenode and link resources compared to other solutions. However,


0

20

40

60

80

100

20 40 60 80 100

Co

mp

uta

tio

n T

ime

(se

co

nds)

Number of Substrate Network Nodes

GNMSP

VNA-I

TANMSP

CNMMCF

PaGeViNE

ViNEOPT

Fig. 12: Average Computation Time

1

4

16

64

256

1,024

4,096

16,384

65,536

GNMSP VNA-I TANMSP CNMMCF PaGeViNE ViNEOPTCom

puta

tion T

ime in

mill

iseconds (Log 2

S

cale

)

Fig. 13: 95% Confidence Interval Error Bars

0

10

20

30

40

50

60

Tota

l Em

bed

din

g C

ost

0

10

20

30

40

50

60

70

0 500 1000 1500

To

tal E

mb

ed

din

g C

ost

Number of Arrivals

GNMSP VNA-I

TANMSP CNMMCF

PaGeViNE ViNEOPT

Fig. 14: Cummulative Embedding Cost

0

30

60

90

120

150

180

0 500 1000 1500

To

tal E

mb

ed

din

g R

eve

nu

e

Number of Arrivals

GNMSP VNA-I

TANMSP CNMMCF

PaGeViNE ViNEOPT

Fig. 15: Cummulative Embedding Revenue

0

20

40

60

80

100

120

0 500 1000 1500

Tota

l E

mbeddin

g P

rofit

Number of Arrivals

GNMSP VNA-I

TANMSP CNMMCF

PaGeViNE ViNEOPT

Fig. 16: Cummulative Embedding Profit

0

20

40

60

80

100

120

1 2 3 4 5

Tota

l E

mbeddin

g P

rofit

Arrival Rate

GNMSP VNA-I

TANMSP CNMMCF

PaGeViNE ViNEOPT

Fig. 17: Effect Arrival Rate on Profit

we note that CNMMCF has a link utilization ratio that iscomparatively close to that of PaGeViNE. Finally, it is evidentfrom Fig. 11 that the utilisation of the resources is almostun affected by the arrival rate. This confirms the fact thatthe rejection of VN requests is not caused by depletion ofresources but rather by inefficient embedding which eitherfails due to bottleneck nodes. This is why mathematicalprogramming-based algorithms which have global knowledgeof the embedding perform better.

2) Solution Complexity: With respect to time complexity,the graphs in Fig. 12 show that the running times of GNMSP,VNA-1 and TANMSP are comparatively lower than those ofPaGeViNE. Once again, this can be explained by the fact thatthese two solutions do not solve a mathematical program asPaGeViNE does. We also note that the computation time ofPaGeViNE is slightly higher than that of CNMMCF. Thiscan be attributed to the fact that PaGeViNE solves threemathematical programs, while CNMMCF solves only two.Moreover, it is expected that solving the problem in one-shotrequires more computation than solving it in two stages, sincesome of the mathematical programs solved in PaGeViNE arebinary. With regard to ViNEOPT we see that the computationtime quickly grows exponentially. In fact, ViNEOPT could notfind a solution even after 1 hour for 60 substrate nodes4. Wetherefore note a significant improvement in time complexityof PaGeViNE compared to ViNEOPT. These simulations wereeach repeated 20 times, and the average time determined ineach case. In Fig. 13, we show the 95% confidence intervals ofthe computation time for a SN with 50 nodes. The small errorvalues in each of these graphs further confirms the profile inFig. 12.

4Once again, this is why the simulations for acceptance ratio were splitinto one with 20 SN nodes and another with 100 substrate nodes.

3) Embedding Cost, Revenue and Profit: Figs. 14, 15and 16 show the cumulative embedding costs, revenue andprofit. The profit is the difference between the revenue andcost of embedding a given VN. We note that PaGeViNEachieves a profitability close that of ViNEOPT, which isconsiderably higher than that of the compared state-of-artapproaches. We also note CNMMCF achieves a higherprofitability than VNA-1, TANMSP and GNMSP. It is worthnoting that these profiles are similar to those obtained fromthe acceptance ratios of the three approaches. This meansthat the superiority of our approach is not based on acceptingVNs with less resources requirements which would be lessprofitable for the physical resource providers. The fact thatVNA-1, TANMSP, GNMSP and CNMCMF obtained muchlower embedding costs is due to rejecting most of the VNrequests, which is further confirmed by the revenue obtained,and hence profitability. In Fig. 17, we evaluated the effect ofthe arrival rate on profitability, noting that as the arrival rateis increased, the profitability reduces. This is not surprisingsince an increase in the arrival rate ensures that most of thearriving VN requests in the simulation time are not accepteddue to lack of resources. This profile is consistent with thatof the acceptance ratio is Fig. 8.

Effect of Initial Solution: In Fig. 18, we evaluate theproposed initial solution. In particular, the effect of the initialsolution on the computation time for a substrate networkof 20 nodes, and the acceptance ratio after the arrival of1,500 VN requests are shown. It can be observed thatthe proposed initial solution achieves the balance betweentime complexity and embedding quality. While it performsworse than GNMSP, VNA-1 and TANMSP in terms ofcomputation time, it outperforms them on solution quality.


Acceptance Ratio after 1500 Arrivals0.260.380.470.610.450.490.550.560.910.700.97

12 48 107

537

1,401 1,4981,632 1,640

1,869

2,856

3,621

0

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

0.00

0.20

0.40

0.60

0.80

1.00

Computation time (milliseconds) for 20 SN Nodes

Acceptance Ratio after 1500 VN Arrivals

Fig. 18: Evaluation of the Initial Solution

80 90 10068 0.9238976 0.923901

62 0.9662 0.9662 56 75,010 92,544 57 75,257 75,257

0

20

40

60

80

100Computation time (Seconds) for 50 SN Nodes

0.84

0.92

1.00

10 20 30 40 50 60 70 80 90 100Number of Iterations

PaGeViNE ViNEOPT

Computation time for 20 SN Nodes

Fig. 19: Effect of Number of Iterations

Even more, it outperforms CNMMCF both on computationtime as well as solution quality. The reason for this slightlybetter performance can be attributed to the fact that inCNMMCF node mapping is finalized by mapping eachvirtual node individually, which could sometimes lead tofailures in embedding especially if more than one virtualnode compete for a given substrate node. We also evaluatedthe performance of PaGeViNE in case the initial solutionis changed. For example, PaGeViNE(GNMSP) means thatGNMSP is used to determine the initial solution beforeapplying path generation. These results show that PaGeViNEis dependent on an initial solution for both solution qualityas well as computation time. With regard to the computationtime, this dependence can be explained by the fact that theinitial solution as well as the main PaGeViNE mathematicalprograms are solved sequentially. This means that if thecomputation of the initial solution takes longer, the overallsolution will take longer. Similarly, since we do not allowthe algorithm to run to completion, the quality of the initialsolution will determine that of the final solution in two ways(1) in some cases, the initial solution just fails to even finda start solution, or (2) if the obtained initial solution is notgood enough, the improvement in one iteration is not asgood as it could be. These aspects are all confirmed by Fig. 18.

Effect of Number of Iterations: Finally, Fig. 19 is aimed atjustifying our decision to perform a single iteration rather thanhaving an iterative approach. From the Fig. we can observethat as the number of iterations is increased, the computationtime increases more rapidly than does the solution quality.

E. LimitationsThe mathematical formulation (1)−(7) involves solving a

binary program. This problem is NP-hard in the general

case, and only exponential algorithms are known to solve itin practice [31]. Our approach is to reduce the number ofinput variables to the program using path generation. Whilea significant improvement in computation time is achievedcompared to the optimal solution, more work can be done forinstance seeking a relaxation to the program which permits tosolve it in polynomial time. We however note that in practicethere are high performance tools [30] for solving binaryprograms. In particular, we have noted that the initial solu-tion also contributes significantly to the overall computationcomplexity, and hence a more efficient heuristic for the samepurpose could possible further enhance the results obtainedin this paper. In addition, it would be interesting to make amathematical analysis on the bounds of the computation timesavings achieved in this paper.

VII. CONCLUSION

In this paper we have proposed a VNE solution whichdiffers from previous solutions by performing node and linkmappings in one shot using optimization theory and pathgeneration. Our path generation based approach first obtainsan initial solution by coordinating the node and link map-ping stages, and then enhances this solution by carrying outonly one round of pricing for the dual variables to obtainthe final solution. Through extensive simulations, we haveshown that our approach has a comparative advantage overprevious approaches in terms of solution quality, achievinga comparatively superior acceptance ratio as well as VNErevenue, which directly leads to higher profitability for SNproviders. The acceptance ratio is atleast 95% of that obtainedby the optimal solution. In addition, our approach significantlyreduces solution computation time compared to the optimalone (achieving a 92% saving in computation time for SNs of


50 nodes), and that this time complexity is comparable to thatof related works.

Looking at the future, there are several possible researchavenues. With regard to time complexity, it would also beinteresting to propose relaxations to the mathematical pro-grams in order to ensure polynomial time convergence. Forthis purpose, we are currently investigating the feasibility ofusing a combination of Tabu Search and path relinking tofurther improve the solution time. In addition, to optimizeresource allocations over time, we are exploring possibilities ofmodeling the substrate network state as a markovian decisionprocess [32], and by assigning state probabilities and transitionrewards be able to bias the mapping of virtual resources tomore appropriate resources. Finally, we intend to extend ourproposed solution to a multi-domain VNE scenario [33] andto consider failures in the SN [34], [35].

ACKNOWLEDGEMENT

The authors are grateful to the editors as well as anonymousreviewers whose insightful comments and suggestions led toa significant improvement in this paper. This work has beensupported in part by FLAMINGO, a Network of Excellenceproject (318488) supported by the European Commission un-der its Seventh Framework Programme and project TEC2012-38574-C02-02 from Ministerio de Economia y Competitivi-dad.

APPENDIXVINEOPT

This is the link based formulation of the one-shot optimalVNE problem. We define f ijuv as the flow of a virtual linklij ∈ Lv on the link luv ∈ (Ls∪Lx). Lx is the set of all metalinks in the augmented SN.

minimize∑

lij∈Lv

∑luv∈(Ls∪Lx)

1

Auvf ijuv +

∑nv∈Nv

∑ns∈Ns

1

Ans

χnvns

subject to

Node Mapping Constraints∑ns∈Ns

χnvns

= 1 ∀nv ∈ Nv

∑nv∈Nv

χnvns≤ 1 ∀ns ∈ Ns

f ijuv −Dijχiu ≤ 0 ∀uv ∈ Lx,∀lij ∈ Lv

f ijuv −Dijχjv ≤ 0 ∀uv ∈ Lx,∀lij ∈ Lv

Capacity Constraints∑ij∈Lv

f ijuv ≤ Auv ∀luv ∈ (Ls ∪ Lx)

∑uv∈Lv

f ijuv = Dij ∀lij ∈ Lv

Flow Conservation Constraints

Source Nodes∑k∈Ns

f ijik −∑k∈Ns

f ijki = Dij ∀lij ∈ Lv

Sink Nodes∑k∈Ns

f ijjk −∑k∈Ns

f ijkj = −Dij ∀lij ∈ Lv

Intermediate Nodes∑u∈Ns

f ijuv −∑u∈Ns

f ijuv = 0 ∀lij ∈ Lv,∀v ∈ Ns

Domain Constraints

f ijuv = [0, Dij ] ∀lij ∈ Lv,∀luv ∈ (Ls ∪ Lx)

χiu = [0, 1] ∀i ∈ Nv,∀u ∈ Ns

REFERENCES

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[11] R. Mijumbi, J.-L. Gorricho, J. Serrat, M. Claeys, F. De Turck, andJ. Famaey, “Neural network-based autonomous allocation of resourcesin virtual networks,” in Proceedings of the European Conference onNetworks and Communications (EuCNC), ser. EuCNC2014, June 2014.

[12] R. Mijumbi, J.-L. Gorricho, J. Serrat, K. Xu, M. Shen, and K. Yang, “Aneuro-fuzzy approach to self-management of virtual network resources,”Journal of Expert Systems With Applications, Feb 2015.

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[34] R. Mijumbi, J.-L. Gorricho, J. Serrat, J. Rubio-Loyola, and R. Aguero,“Survivability-oriented negotiation algorithms for multi-domain virtualnetworks,” in Network and Service Management (CNSM), 2014 10thInternational Conference on, Nov 2014, pp. 276–279.

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Rashid Mijumbi obtained a Bachelors of ScienceDegree in Electrical Engineering from MakerereUniversity (Kampala, Uganda) in 2009, and a PhDin Telecommunications Engineering from the Uni-versitat Politecnica de Catalunya (UPC) (Barcelona,Spain) in 2014. He is currently a Postdoctoral Re-searcher in the Network Engineering Department atUPC. His research interests are in management ofnetworks and services for the future Internet. Currentfocus is on resource management in virtualizednetworks and functions, software defined networks

and cloud computing.

Joan Serrat received a degree of telecommunicationengineering in 1977 and a PhD in the same fieldin 1983, both from UPC. Currently, he is a fullprofessor at UPC where he has been involved in sev-eral collaborative projects with different Europeanresearch groups, both through bilateral agreementsor through participation in European funded projects.His topics of interest are in the field of autonomicnetworking and service and network management.He is the contact point of the TM Forum at UPC.

Juan-Luis Gorricho received a telecommunicationengineering degree in 1993, and a Ph.D. degree in1998, both of them from the Technical Universityof Catalonia (UPC). Since 1994 he joined the De-partment of Network Engineering at the UPC asan assistant professor, and as associate professorsince 2001. His most recent research interests havebeen focused on applying artificial intelligence to theresearch fields of ubiquitous computing and networkmanagement; with special interest on using smart-phones to achieve the recognition of user activities

and locations; and applying linear programming and reinforcement learningto solve the network embedding problem and the resource allocation problem,targeting the implementation of the network virtualization and the futurenetwork function virtualization.

Raouf Boutaba received the MSc and PhD degreesin computer science from the Universite de Pierreet Marie Curie, Paris, France, in 1990 and 1994,respectively. He is currently a full professor ofcomputer science at the University of Waterloo, Wa-terloo, ON, Canada, and a distinguished visiting pro-fessor at the Pohang University of Science and Tech-nology (POSTECH), Korea. His research interestsinclude network, resource and service managementin wired and wireless networks. He has receivedseveral best paper awards and other recognitions

such as the Premier’s Research Excellence Award, the IEEE Hal Sobol Awardin 2007, the Fred W. Ellersick Prize in 2008, the Joe LociCero and the DanStokesbury awards in 2009, and the Salah Aidarous Award in 2012. He is afellow of the IEEE and the Engineering Institute of Canada.

http://dl.acm.org/citation.cfm?id=241938.241942


www.cs.columbia.edu/coms6998-3/lpprimer.pdf?

www.cs.columbia.edu/coms6998-3/lpprimer.pdf?

http://coral.ie.lehigh.edu/~ted/files/papers/Duality-EOR10.pdf

http://coral.ie.lehigh.edu/~ted/files/papers/Duality-EOR10.pdf

http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/about/

http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/about/


http://dx.doi.org/10.1186/1869-0238-4-6

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