A Multi-Objective Approach for VNE Problems using multiple ILP formulations

Network Virtualization is a key technology for the Future Internet, allowing the deployment of multiple independent virtual networks that use resources of the same basic infrastructure. An important challenge in the dynamic provision of virtual networks resides in the optimal allocation of physical resources (nodes and links) to requirements of virtual networks. This problem is known as Virtual Network Embedding (VNE). For the resolution of this problem, previous research has focused on designing algorithms based on the optimization of a single objective. On the contrary, in this work we present a multi-objective algorithm, called VNEMO-ILP, for solving dynamic VNE problem, which calculates an approximation of the Pareto Front considering simultaneously resource utilization and load balancing. Experimental results show evidences that the proposed algorithm is better or at least comparable to a state-of-the-art algorithm. Two performance metrics were simultaneously evaluated: (i) Virtual Network Request Acceptance Ratio and (ii) Revenue/Cost Relation. The size of test networks used in the experiments shows that the proposed algorithm scales well in execution times, for networks of 84 nodes.


Introduction
Network Virtualization is an important component in the evolution of Internet since it enables functions that were not available in the original design, and it breaks with the process of Internet ossification [1,2].Besides, many testbeds of academic and industry organizations are based on this paradigm with the goal of providing an independent research environment to different working groups in the area [3][4][5].
A virtual network is formed by virtual nodes and virtual links, and shares resources from nodes and links of the physical or substrate network.Both virtual nodes and virtual links form a Virtual Topology [6].Multiple virtual networks can coexist, isolated and independent from each other, mapped on the same physical hardware.
The problem of assigning physical network resources to Virtual Networks Requirements (VNR) is known as VNE (Virtual Network Embedding) and it is critical for the deployment of Network Virtualization [7].As in most engineering optimization problems, the goal is to maximize benefits with an efficient allocation.These benefits can refer to maximizing profits, optimal utilization of physical resources or achieving a desired QoS (Quality of Service).VNE problem is intrinsically complex, not only in its mathematical formulation but also at the computational level [8], and currently it is considered as a NP-hard problem [6].
Due to its complexity, VNE is usually divided in 2 sub-problems: (1) Node Mapping, that assigns resources (as processing capacity) of physical nodes to virtual nodes, and (2) Virtual Link Mapping that assigns links or path formed by consecutive physical links with their corresponding bandwidth, to virtual links.Virtual nodes only can be assigned to physical nodes if there are sufficient resources for hosting this virtual node in the physical node.Analogously, virtual links have to be assigned to physical links with available resources (as bandwidth) to host them.
We can consider two versions of the VNE problem: the static or offline version, in which VNRs are known in advance, and the dynamic or online version, where the VNR are treated individually, one at a time, as they appear in a time basis.This work deals with the online version.
Recently, VNE problem has attracted a lot of attention from the scientific community, as we can see in published surveys [6,9].Proposed methods for the VNE resolution are mainly based on heuristics and ILP (Integer Lineal Programming) formulations.
Early works dealing with VNE have proposed algorithms in which the allocation of virtual nodes and links are made in an independent way, with no coordination between these two sub-problems, as in [7,10].On the other hand, Chowdhury et al. proposed coordinated allocation of virtual nodes and virtual links, but in two separated phases [11].
In [12], Cheng et al. performed both allocations in one coordinated phase, using topological attributes of the VNR and the Substrate Network, elaborating a ranking of nodes.This work was inspired in the PageRank algorithm used by Google for web page ranking [13].In this way, they tried to take into account the incidence of the network topology.In the same line, the authors of [14] present a heuristic that maps virtual nodes and virtual links in a coordinated way.A new metric is proposed, named Global Resource Capacity (GRC), which takes into consideration the network topology.Based on GRC, the heuristic first maps all virtual nodes, and then performs virtual link mapping based on shortest paths.The objective is to minimize the revenue-cost relation.
The work of Lischka and Karl in [15] also makes the allocations of nodes and links in a coordinated way, in this case using Subgraph Isomorphism Detection (SID) algorithms.With this proposal, an isomorph subgraph that represents a VNR is searched in the physical topology, applying restrictions that limit the quantity of hops to map the virtual links.
In [16] the authors make the observation that in many real scenarios, resources requirement of most applications vary in time.They present a model that exploits this situation and try to utilize unused resources of virtual networks to share in a more efficient way physical network resources.However, this strategy breaks with the isolation that must exist among virtual entities sharing the same substrate.As a final example of heuristics applied to VNE, [17] analyzes physical topology in order to plan physical network expansion, identifying partitions and cut links, looking for a better interconnection of the network while avoiding blocking of requirements.
All these cited works solved the VNE problem with heuristics that optimize a single specific objective.In [8], Melo et al. used an ILP formulation getting good results considering performance metrics as well as running time.They have evaluated three different objective-functions for the "node-link formulation", concluding that the best option was the Weighted Shorted Distance Path (WSPD), which combines weighted objectives of load balance and shortest paths for the virtual links.A later work using exact methods is [18] which uses column generation strategies in order to minimize computational load and facilitate the scalability of the solution to a dynamic VNE problem.
While all these approaches are applied to generic packet-switched physical networks, some other works focus on a single technology in the substrate network for the purpose of establishing specific restrictions and proposing a solution over a real scenario.For optical networks, late works are centered on Elastic Optical Networks (EON), in which the optical fiber spectrum is divided in Frequency Slots (FS).One or more contiguous FS can be used by a transducer, depending on its data bit rate and modulation technique.The work presented in [19] presents an exact solution for Virtual Optical Network Embedding (VONE) problem, based on the concept called Maximum contiguous slot-block (MCSB).Besides, they proposed two heuristics for both transparent (with no electrical-optical-electrical conversion in physical nodes) networks and opaque networks.Other work in the same area is [20] which treats multicast traffic in virtual networks, over optical EON physical substrates.For wireless networks, an example is [21] where the authors propose the use of partially available spectrum bands as aggregated capacities in 3G or LTE technologies.
The present paper is an extension of our previous work [22] in which we presented a multi-objective approach based on the idea that a trade-off exists between the cost of the allocation in terms of the utilization of physical resources, and the balance in a uniform distribution of resources in the substrate network, two of the most studied objective functions and therefore, both objectives should be optimized simultaneously.To the best of our knowledge, this work was the first in proposing a multi-objective approach that allows the calculation of a Pareto Front approximation [23] for the VNE, considering simultaneously the two above cited objective functions.In our previous work, we presented limited experiments using two different medium size network topologies.In what follows, the experiments were extended here to network topologies up to 84 nodes and 102 links, which can be already considered as a large network.Experimental results will be presented to demonstrate that the proposed algorithm scales well enough even to these large networks, confirming previous results published in [22].
The proposed algorithm, named in this work VNE-MO-ILP (Virtual Network Embedding -Multi-objective -Integer Lineal Programming) performs multiple executions of an ILP formulation, each one with variations on a given restriction.In this way, the Pareto Front is generated one point at a time, and the network operator has multiple tradeoff options for allocating resources when allocating resources to a VNR.
Since the network operator has multiple options for the allocation of a single VNR, it is also interesting to investigate the effect of the criterion to be followed in the selection of a specific Pareto solution at each time instant, over the global performance metrics usually used for evaluating the efficiency of the algorithm, as: (1) the VNR Acceptance Ratio, and (2) the Revenue/Cost Relation [12].
The remainder of this work is organized as follows.In Section II, we formulate a formal description of the VNE problem.In Section III, the proposed algorithm VNE-MO-ILP is described.The simulation results, using a network simulator, is presented in Section IV.Finally, Sections V and VI conclude this paper.

Formal Description of the VNE Problem
This section describes the modeling of virtual and physical networks, defining also main restrictions in VNE problem formulation.This formulation is based on the proposal of Melo et al. [8].We also present the performance metrics that will be used to evaluate the efficiency of the proposed algorithm, when compared to a state of the art alternative [8].

Modeling of the physical network and the VNR
The physical (or substrate) network is modeled as a weighted undirected graph   = {   ,   ,   ,   ,   } composed by a set of physical nodes   and a set of physical links   .Each physical node  ∈   is characterized by a processing capacity    (for instance, CPU units).We consider that each physical link   ∈   has a bandwidth    and a specific propagation delay    .
In turn, a single VNR can be described as a weighted undirected graph   = {   ,   ,   ,   ,   } composed by a set of virtual nodes   and a set of virtual links   .Each virtual node  ∈   is characterized by the requirement of some processing capacity    .With respect to virtual links, each   ∈   ,  < , requires a bandwidth    with a maximum propagation delay of    .This work deals with the dynamic problem, so each VNR is known in a specific time t I and has a life time   after which, it must be retired from the physical network and the used physical resources can be released for latter reuse by another VNR.
Besides,    represents the sub-set of physical links   that are directly connected to physical node i in the physical network.Similarly,    represents the subset of virtual links   that are directly connected to virtual node  in a given VNR.

Mapping Variables
The binary variable    represents the mapping of virtual nodes to physical nodes and it is defined by the expression: The binary variable    represents the allocation of a virtual link to a path formed by consecutive physical links: After a successful allocation and before a new incoming VNR is considered, the physical network must be updated, subtracting the resources in use by the new virtual network from the potentially available physical links and nodes.

VNE problem Restrictions
In order to ensure a correct mapping of virtual nodes and links, and obeying the conservation of physical network resources law, a set of restrictions are defined following the guideline proposed in [8] by Melo et al.

Allocation of virtual nodes to physical nodes
Relations (3) to (5) ensure that each virtual node is assigned to only one physical node, and that each physical node can be used by at most one virtual node per each virtual network (in the same VNR).Besides, the capacity of each physical node cannot be exceeded:

Allocation of virtual links to physical links
In order to allow simultaneous optimization in the allocation of virtual links and virtual nodes, the Multi-Commodity Flow [24] restriction is applied as a whole, considering the Node-Link Formulation [8].Besides, the notion of directional flows is used over the virtual links that represent expression (6), following the notation proposed in [8].

Bandwidth Limitation
Expression (7) assures that the available bandwidth capacity at each physical link will not be exceeded.

Limit in the Propagation Delay of virtual links
The maximum delay    of a virtual link lmn is a parameter of the problem that narrows the value of the additional physical link's delay that composes a virtual link.Melo et al. apply (8) to assure that this delay restriction is fulfilled:

Performance metrics for VNE problem
It is necessary to define some metrics to evaluate the efficiency of the studied mapping algorithm.In the case of the dynamic version of the VNE problem, i.e. when VNRs arrive at different time slots without a priori knowledge, these metrics are calculated at the end of the simulation process, after making all allocations of virtual networks in a time basis.
The two performances metrics that we will use in this work are the most accepted in the specialized literature [10,11] and are the following:

VNR Acceptance Ratio
This metric is denoted as   , and it is given by equation (9).It measures the global performance of the allocation method considering its capacity of mapping (and not blocking or rejecting) as many VNRs as possible.
In this equation, k VNR represents the total number of arriving VNRs, while k VNRa is the number of VNRs successfully satisfied by the studied algorithm.

Revenue / Cost Relation
The cost of a VNR mapping is considered as the cost of the resources in the physical or substrate network that were used in the mapping process.The revenue can be considered as proportional to the costs of resources requested by a single VNR (CPU of virtual nodes and Bandwidth of virtual links).This indicator E  gives an idea of the efficiency in the allocation process for the use of network resources and therefore, it represents more profit for the owner of the network infrastructure [7].Coefficients α and β relate real costs of capacity units in the nodes to link bandwidth costs.

VNE-MO-ILP Algorithm
When an algorithm assigns resources to a given VNR, the objective of this assignation should point to get good results with the considered global performance metrics, in this case, Acceptance Ratio A VNR and Revenue/Cost Relation E VNR .
It is necessary to clearly understand the difference between the Global performance metrics and the Objectives for a single VNR.Global performance metrics evaluate the entire VNE process, i.e. at the end of all VNR assignment attempts, and measures the efficiency of the VNE algorithm as a whole.For an online process, they are calculated at the end of a large period of time, after many VNR allocations were made.For an offline process, the global performance metrics are calculated when all VNRs were attempted to allocate.These global metrics can be seen as a posteriori metrics.On the other hand, the objectives for a single VNR are optimization objectives applied to the less complex problem of the allocation of each VNR, i.e., they are a priori metrics.
This work proposes for the first time the simultaneous optimization of both cited objectives at each mapping, instead of considering only one objective at a time, as already proposed in previous works, i.e., our proposal will simultaneously try to:  Minimize the utilization of physical resources: the requirement of virtual nodes cannot be minimized, so it is intended to minimize the cost of used physical links, in order to satisfy the requirements with the least quantity of resources as possible, looking for efficiency.This objective has a direct relation with metric E VNR ; and  Maximize load balance: in this case, the physical network resources will be assigned in such a way that the network remains with balanced resource availability in the nodes and links, in order to facilitate subsequent allocations and avoiding isolation of resources.In this way, we will try to improve the Acceptance Ratio A VNR metric.
In the wake of the above descripted ideas, the algorithm proposed in this work, named here VNE-MO-ILP (Virtual Network Embedding-Multi-objective-Integer Lineal Program-ming) will:  Define a mapping cost SP, given by equation (11), where SP is limited for a defined a priori minimum and maximum values.The algorithm takes (s+1) possible values of SP, equally spaced between maximum and minimum values, as will be shown in equation (15).(11)  Optimize Load Balance: the algorithm looks for resources in physical nodes with more available capacity, and in physical links with more bandwidth; therefore, the objective function of each ILP formulation will be (12), where  defines a relation between the costs of physical nodes and links.
The values of maximum load in physical nodes and links, ℒ    and ℒ    respectively, are defined according to the following new restrictions which are added to the former restrictions already formulated in Section 2: The proposed VNE-MO-ILP algorithm calculates an approximation of the Pareto Front, using the above summarized ideas.To the best of our knowledge, no work previous to [22] proposed a multi-objective approach to the VNE problem, even when more than a single performance metric to evaluate the efficiency of VNE algorithms is used (usually, as a weighted sum of different objectives).
In this way, the proposed algorithm offers to a network operator a whole Pareto Front approximation with different solutions (all optimal in a Pareto sense [23]) for a single mapping, in order to allow him to choose a Pareto optimal solution that best fits his requirements, in a specific situation, at a given moment in time.
The Algorithm VNE-MO-ILP presented in Table 1, initializes variables and parameters, then calculates the acceptable limits of parameter SP defined in (11), and the interval [SPmin, SPmax] is divided in s sub-intervals, to later assign consecutively those (s+1) values to the SP parameter: Then, an ILP formulation is built with the following elements:  Parameters, assignation variables and restrictions presented in Section 2;  Additional restrictions of expressions (11), ( 13) and ( 14);  The objective-function defined in (12).This way, an ILP solver, which solves mono-objective problems, is used to find an approximation of the Pareto Front, being this Pareto Front a typical result of a pure multi-objective problem [23].
Although ILP method is not scalable to large instances of any problem, our proposed algorithm uses anyway this tool, taking into account the very good execution times reported in the reference work [8], and being aware that for more complex networks, efficient heuristics or meta-heuristics will be needed.
In this paper we take three different network topologies, which are representative of medium and large sizes ISP (Internet Service Provider) networks, with very good results and reasonable execution times, proving the viability of this proposal, leaving for future work the development of efficient metaheuristics for the VNE problem in a pure multiobjective context.
Table 2 highlights main differences between the algorithms proposed in this work VNE-MO-ILP with the VNE-NLF algorithm presented in [8], which is taken as a reference to evaluate the presented experimental results.

Discrete-Event Network Simulator
We have developed a discrete-event Network Simulator in Java language for the evaluation of the proposed algorithm, which is outlined in Figure 1.This simulation tool covers exactly all restrictions of the proposed VNE problem, as we presented in Section III, and permits the execution of a complete instance of the VNE problem (a set of VNR stored on a database, treated one by one in a time basis).This network simulator is available for the scientific community at: http://www.cc.pol.una.py/VNE-MO-ILP/.
After defining parameters of the physical network and a set of M virtual network requirements (VNR), the simulator randomly takes a VNR and it passes this requested VNR to the proposed VNE-MO-ILP algorithm, which interacts with the ILP Solver to build a Pareto Front approximation in a point-to-point basis with each calculated non-dominated solution (optimal in the Pareto sense).If the algorithm could not find any solution, the performance metrics are updated.On the contrary, if a Pareto Front approximation is found with more than a single solution, it is necessary to apply a (possibly subjective) criterion to choose a single solution, to update performance metrics and network parameters and to pass to the next time slot to continue with the simulation.

Test Networks and experimental environment
Three substrate networks were used in the following reported experiments: IRIS, MARNET, and BESTEL.These topologies were obtained from The Internet Topology Zoo [25] and are shown in Figures 2, 3 and 4 respectively.IRIS is a mesh topology with 51 nodes and 64 links; MARNET is a hybrid topology (star/mesh) with 20 nodes and 27 links.Both are representative of medium size networks.The BESTEL mesh topology has 84 nodes and 102 links and it is a typical large size network.The propagation delay at each physical link is proportional to its length.For the reported simulations, capacities at each node were generated randomly with values ranging from 200 to 300 capacity processing units.The transmission rate at each link is considered to be 512 Mbps.
The VNRs were generated using network topology generator BRITE [26].The number of virtual nodes varies between 2 and 10 nodes.The virtual link bandwidth requirement takes an integer value uniformly distributed between 2 and 8 Mbps at each link, while the requirements in virtual nodes were uniformly distributed between 2 and 40 capacity processing units.
For the resolution of the ILP sub-problems, solver IBM Ilog CPLEX ® [27] version 12.6 was used.The execution times in the tests were registered for later comparisons.

Experimental Results
We performed groups of experiments taking VNRs with average life time values of: 10, 20, and so on, until 100 time units.We assumed the arrival of a new VNR at each time slot.The experiments are evaluated after 500, 1000, and so on until 5000 discrete time slots, allowing the network to reach its stable state regarding the number of installed VNRs.All simulations were executed in machines with Intel Core (TM) I7-4770 (3.4 GHz) with a RAM of 8 GB.
As was mentioned early, this paper presents a multi-objective approach for the first time in VNE literature.Therefore, a fair comparison with any previous work is difficult to develop.For comparison purposes, we have implemented the VNE-NLF algorithm presented in [8] in our simulation tool, and made an execution of the same instances of the problem.VNE-NLF algorithm was chosen as representative of the state of the art considering its great performance in terms of the same metrics used in this work, and with reasonable execution times.Besides, the VNE formulation used in this work is the same as the one used in [8].

Solution selection Criteria
For experimental purposes, we have considered three possible criteria for the selection of a unique solution from a Pareto Front approximation, in order to assign resources to the considered VNR at each discrete time interval, using this way the selected solution to simulate the situation at the next time slot.
1) The first criterion is to take the solution of the Pareto Front with the least Assignation Cost SP (11), and worst Load Balance (12) which is called Left Criterion (LC).
2) For the second criterion, we take the nearest solution to origin of coordinate axis, looking for a trade-off between both objectives.This criterion will be called Central Criterion (CC).
3) Last, we take the solution of the Pareto Front with the largest Assignation Cost but with the best Load Balance (Right Criterion, RC). Figure 5 shows an example of a Pareto Front, corresponding to a single VNR assignation, in which the three criteria of selection are indicated.The obtained results, comparing the 3 proposed Selection Criteria, are shown in Tables 3, 4 and 5 for IRIS, MARNET and BESTEL networks respectively.Presented values are average values of five runs for each instance.It is clear from the presented Tables that the proposed VNE-MO-ILP algorithm outperforms the reference algorithm VNE-NLF presented in [8] in almost all comparisons, except for the values marked in bold and red in Table 4 (MARNET topology), for Lifetimes of 60, 70 and 90.In fact, Tables 3 (IRIS topology) and 5 (BESTEL topology) show that our proposed algorithms systematically outperformed the reference algorithm proposed by Melo et al. in all the experiments when simultaneously considering both performance metrics: Acceptance Ratio (A VNR ) and Revenue/Cost Relation E VNR .
Even when considering the experimental results presented in Table 3 for the hybrid MARNET network, the best values of Revenue/ Cost Relation E VNR (considered alone) were reached by the proposed VNE-MO-ILP algorithm, while the reference algorithm VNE-NLF only outperforms VNE-MO-ILP in 3 out of 10 instances.In other words, VNE-MO-ILP finds solutions that dominate in the Pareto sense the solutions calculated by the NVE-NLF algorithm while the latter is not able to find even one solution that dominates the ones calculated by the proposed algorithm.
In short, solutions calculated by the proposed VNE-MO-ILP algorithm are systematically non-dominated (Pareto optimal) and they dominate most solutions calculated with the state of the art algorithm of Melo et al.On the contrary, no solution calculated by this state of the art VNE-NLF algorithm is able to dominate (in the Pareto sense) even a single solution calculated with the proposed algorithm, proving this way the advantages of using the VNE-MO-ILP algorithm.

Revenue/Cost Relation
Figures 6, 7 and 8 present experimental results considering this E VNR metric for each of the 3 considered topology as a function of VNR's lifetime, which varies from 10 to 100 time units.Those three figures show that as the lifetime average increases, a larger number of virtual networks remains simultaneously installed, and more physical network resources are utilized.
For the IRIS network (Fig. 6), the three selection criteria outperform the results obtained by the reference algorithm.The best selection criterion is LC (Left Criterion) as expected, given that it mainly optimizes resources utilization in the mapping of each VNR.
For the MARNET network (Fig. 7), the results using VNE-MO-ILP are also better than the ones using the reference algorithm.However, this time only two criteria obtain better results than the reference algorithm: the Left Criterion (LC) and the Central Criterion (CC).For this experiment, the Right Criterion (RC) only obtains a performance similar to the reference algorithm without significant improvement, as it was the case with the IRIS network shown in Fig. 6.This difference may be explained taking into account that the VNE-NLF algorithm considers, in a single ILP formulation, both components: load balance and resource utilization, without looking at the trade-off between both objectives.On the contrary, proposed VNE-MO-ILP algorithm considers both aspects separately, calculating a Pareto Front approximation, which characterizes this trade-off, achieving in this way better results.
For the BESTEL network, the results presented in Figure 8 show that they are very similar to the experimental results for the IRIS network (Fig. 6).The three selection criteria outperform the reference algorithm.In particular, the LC criterion is once more the best option, reaching values of near 80% for E VNR .In fact, VNE-NLF begins with very low values, getting better with more loaded instances, until reaching an E VNR of only 50%, while the other two selection criteria (CC and RC) get very similar results, in the range of 50-60%.

VNR Acceptance Ratio
Figures 9, 10 and 11 show experimental results for this metric.It can be noticed that for the IRIS network (Fig. 9) the proposed algorithm successfully assigns almost 100% of the VNRs, with all 3 selected criteria (LC, CC, RC).However, the reference algorithm only gets up to 95% to 97%, without attending all VNRs.On the contrary, for the MARNET network (Fig. 10) there is no significant difference among the 4 compared algorithms, i.e., the performance of all algorithms are very similar.
For the BESTEL network (see Fig. 11), the proposed algorithm consistently gets 100% in all instances, probably because BESTEL is a large mesh network and the tests could not saturate the network reaching to the limits where we can observe some blocked requests.On the contrary, the reference algorithm could not avoid some blocked requests for loaded instances, proving once more the advantage of using the proposed VNE-MO-ILP algorithm.
These successful results for the proposed VNE-MO-ILP algorithm may be explained considering that it carries out several attempts for mapping each VNR, achieving this way a better search.While some of these candidate solutions may not be feasible, the algorithm tries with almost all possible values of the parameter SP, accomplishing an efficient mapping of almost all VNRs.On the contrary, the reference algorithm only attempts to find a solution once; therefore, it blocks the considered VNR if a solution is not found in this single attempt.
The MARNET network is a star/mesh topology, so it does not offer many options at the time of a blocked request; therefore, the results are very similar with both algorithms, as shown in Fig. 11.

Execution Times
Average execution times for the allocation of a single VNR for the VNE-MO-ILP algorithm took values between two and five seconds while for the VNE-NLF algorithm took about one second, given that it only needs to solve one ILP for each VNR.Logically, it is faster than the VNE-MO-ILP that needs several calls to the ILP Solver, consequently needing a shorter running time.

Conclusions and Future Works
The VNE problem deals with the efficient allocation of resources from a physical network (nodes and links) to virtual network requirements.In this work, the algorithm VNE-MO-ILP (Virtual Network Embedding-Multi Objective-Integer Lineal Programming) is proposed for the resolution of the VNE problem in a dynamic (or online) context, using a multi-objective approach.
The objectives usually considered for the allocation of a single VNR are two: (i) Allocation cost, which is related to the efficient utilization of physical resources, and (ii) Load Balance, that looks for a uniform distribution of resources in nodes and links of a physical network.
The VNE-MO-ILP proposed algorithm obtains an approximation of the Pareto Front for each single requirement, through multiple executions of ILP formulations, finding trade-off alternative solutions between the two considered objectives: (1) utilization of physical links and (2) load balancing.By calculating a Pareto front approximation, this algorithm gives the network operator several options for selecting a specific solution.It is also worth mentioning that this work is an extension of [22], which was, in the best of our knowledge, the first work in proposing a multi-objective approach for the VNE problem.
Many experimental tests were performed comparing the efficiency of the proposed algorithm to a state of the art algorithm as VNE-NLF [8], considering three network topologies of medium and large sizes.Presented experimental results clearly prove that the proposed algorithm systematically outperforms the reference algorithm in the two considered a posteriori metrics: (i) VNR Acceptance Ratio, and (ii) Revenue / Cost Relation, with reasonable execution time.
As this work highlights, the solutions found by the proposed VNE-MO-ILP algorithm dominate in most cases the solutions calculated with the reference algorithm.In other words, the solutions found by the VNE-MO-ILP algorithm are better (in the Pareto sense) than the ones calculated with the reference algorithm, considering simultaneously both performance metrics.We emphasize the relevant experimental fact that no solution calculated with the proposed algorithm was dominated by any solution on the reference state of the art algorithm.
It is worth remembering that several experiments were performed with three different automatic criteria for the selection of a single solution from each Pareto Front approximation (Left Criterion -LC, Center Criterion -CC and Right Criterion -RC).Experimental results indicate that the Left Criterion-LC, which uses the solution with the best physical resource utilization, achieved the best experimental results.This does not necessarily mean that this is the only possible option, since the network operator could change the chosen option at each stage of the process, using different criteria at different moments, depending on the specific needs at each decision time.Logically, any solution of a Pareto Front cannot be considered a bad decision, given that it is a non-dominated solution (optimal in a Pareto sense); however, in the long run, one choice may result better than another, depending on the dynamics of the VNR requirements.
As it was already said, the utilization of ILP formulations in large networks can be prohibitive because of its execution time.However, in this work we presented experimental results with a network of 84 nodes and 102 links (BESTEL network of Fig. 4) with reasonable execution times.For larger networks, the authors propose the utilization of different multi-objective meta-heuristics for the VNE resolution, as ACO (Ant Colony Systems) and PSO (Particle

Figure 5 :
Figure 5: Considered selection criteria from a Pareto Front, for a single VNR allocation, including Left, Central and Right criteria.

Figure 11 :
Figure 11: Results -VNR Acceptance Ratio, BESTEL network Furthermore, in this work we extend the experiments to a large network (BESTEL topology), verifying experimentally that the execution times still remain acceptable.In any case, we propose as future work the development of new efficient algorithms as ACO (Ant Colony Systems), Evolutionary Algorithms (EA), Harmony Search (HS) or PSO (Particle Swarm Optimization), for networks with more than 100 nodes.

Table 1 :
VNE-MO-ILP AlgorithmSubsequently, at each iteration, an ILP Solver tries to solve this ILP problem, returning a solution, if it exists.Each solution calculated by the ILP Solver is a point of the Pareto Front approximation, which will be the output of the VNE-MO-ILP algorithm.
Input: Updated physical network G, requirement VNR, ILP formulation, number of ILP executions s, Max_VNR_Cost.