An efficient linear programming rounding-and-refinement algorithm for large-scale network slicing problem
aa r X i v : . [ c s . N I] F e b AN EFFICIENT LINEAR PROGRAMMING ROUNDING-AND-REFINEMENT ALGORITHMFOR LARGE-SCALE NETWORK SLICING PROBLEM
Wei-Kun Chen ⋆ , Ya-Feng Liu † , Yu-Hong Dai † , and Zhi-Quan Luo ‡ ⋆ School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China † LSEC, ICMSEC, AMSS, Chinese Academy of Sciences, Beijing, China ‡ Shenzhen Research Institute of Big Data and The Chinese University of Hong Kong, Shenzhen, ChinaEmail: [email protected], {yafliu, dyh}@lsec.cc.ac.cn, [email protected]
ABSTRACT
In this paper, we consider the network slicing problem which at-tempts to map multiple customized virtual network requests (alsocalled services) to a common shared network infrastructure and al-locate network resources to meet diverse service requirements, andpropose an efficient two-stage algorithm for solving this NP-hardproblem. In the first stage, the proposed algorithm uses an iterativelinear programming (LP) rounding procedure to place the virtual net-work functions of all services into cloud nodes while taking trafficrouting of all services into consideration; in the second stage, theproposed algorithm uses an iterative LP refinement procedure to ob-tain a solution for traffic routing of all services with their end-to-enddelay constraints being satisfied. Compared with the existing algo-rithms which either have an exponential complexity or return a low-quality solution, our proposed algorithm achieves a better trade-offbetween solution quality and computational complexity. In particu-lar, the worst-case complexity of our proposed algorithm is polyno-mial, which makes it suitable for solving large-scale problems. Nu-merical results demonstrate the effectiveness and efficiency of ourproposed algorithm.
Index Terms — LP Relaxation, Network Slicing, Resource Al-location, Rounding-and-Refinement.
1. INTRODUCTION
Network function virtualization (NFV) plays a crucial role in thefifth generation (5G) and beyond 5G networks [1]. Different fromtraditional networks where service functions are processed by spe-cialized hardwares in fixed locations, NFV efficiently takes theadvantage of cloud technologies to configure some specific nodes(called cloud nodes) in the network to process network service func-tions on-demand, and then flexibly establishes a customized virtualnetwork for each service request. However, as virtual networkfunctions (VNFs) of all services run over a shared common net-work infrastructure, it is crucial to allocate network (e.g., cloud andcommunication) resources to meet the diverse service requirements.The above resource allocation problem in the NFV-enabled net-work is called network slicing in the literature. Various approacheshave been proposed to solve it or its variants; see [2]-[18] and thereferences therein. These approaches can generally be classified intotwo categories: (i) exact algorithms that solve the problem to globaloptimality and (ii) heuristic algorithms that aim to quickly find a fea-sible solution for the problem. In particular, references [2]-[6] pro-
Ya-Feng Liu is the corresponding author. posed the mixed integer linear programming (MILP) formulationsfor the network slicing problem and used standard MILP solverslike Gurobi [19] to solve their problem formulations. References[7]-[10] proposed a column generation approach [20] to solve therelated problems. Though the above two approaches can solve thenetwork slicing problem to global optimality, they generally sufferfrom low computational efficiency as their worst-case complexitiesare exponential. Due to this, references [11]-[18] simplified the so-lution approach by decomposing the network slicing problem intoa VNF placement subproblem (which maps VNFs into cloud nodesin the network) and a traffic routing subproblem (which finds pathsconnecting two adjacent VNFs in the network) and solving each sub-problem separately. To obtain a binary solution for the VNF place-ment subproblem, references [11, 12] first solved the linear program-ming (LP) relaxation of the network slicing problem and then useda rounding strategy while references [13]-[18] used some greedyheuristics (without solving any LP). Once the VNFs are mapped tothe cloud nodes, the traffic routing subproblem is solved by usingshortest path, k -shortest path, or multicommodity flow algorithms.However, solving the VNF placement subproblem without taking theglobal information (i.e., traffic routing of all services) into accountcan lead to infeasibility or low-quality solutions. Therefore, algo-rithms that find a high-quality solution of the network slicing prob-lem while still enjoy a polynomial-time complexity are still highlyneeded.In this paper, we propose a two-stage LP rounding-and-refinementalgorithm which achieves a good trade-off between high solutionquality and low computational complexity. Specifically, in the firststage, we solve the VNF placement subproblem by using an iterativeLP rounding procedure, which takes traffic routing into account; inthe second stage, we solve the traffic routing subproblem by using aniterative LP refinement procedure to find a solution that satisfies theend-to-end (E2E) delay constraints of all services. In particular, theproposed algorithm has a guaranteed polynomial-time worst-casecomplexity, and thus is particularly suitable for solving large-scaleproblems. Numerical results demonstrate the effectiveness andefficiency of our proposed algorithm over the existing ones.
2. SYSTEM MODEL AND PROBLEM FORMULATION
Let G = {I , L} be the directed network, where I = { i } and L = { ( i, j ) } are the sets of nodes and links, respectively. Eachlink ( i, j ) has an expected (communication) delay d ij [14, 17, 18],and a total data rate upper bounded by the capacity C ij . The set ofcloud nodes is denoted as V ⊆ I . Each cloud node v has a computa-ional capacity µ v and processing one unit of data rate requires oneunit of (normalized) computational capacity, as assumed in [11]. Aset of flows K = { k } is required to be supported by the network.The source and destination nodes of flow k are denoted as S ( k ) and D ( k ) , respectively, with S ( k ) , D ( k ) / ∈ V . Each flow k relates toa customized service, which is given by a service function chain(SFC) consisting of ℓ k service functions that have to be processedin sequence by the network: f k → f k → · · · → f kℓ k [21, 22, 23].To minimize the coordination overhead, each function must be pro-cessed at exactly one cloud node, as required in [4, 11, 14]. If func-tion f ks , s ∈ F ( k ) := { , . . . , ℓ k } , is processed by cloud node v in V , the expected NFV delay is assumed to be known as d v,s ( k ) ,which includes both processing and queuing delays [14, 18]. Forflow k , the service function rates before receiving any function andafter receiving function f ks are denoted as λ ( k ) and λ s ( k ) , respec-tively. Each flow k has an E2E delay requirement, denoted as Θ k .The network slicing problem is to determine functional instanti-ation, the routes, and the associated data rates on the correspondingroutes of all flows while satisfying the capacity constraints on allcloud nodes and links, the SFC requirements, and the E2E delay re-quirements of all flows. Next, we shall briefly introduce the problemformulation; see more details in [2]. • VNF Placement
We introduce the binary variable x v,s ( k ) to indicate whether ornot function f ks is processed by cloud node v . Each function f ks mustbe processed by exactly one cloud node, i.e., X v ∈V x v,s ( k ) = 1 , ∀ k ∈ K , ∀ s ∈ F ( k ) . (1)Let y v = 1 denote that cloud node v is activated and powered on;otherwise y v = 0 . Thus x v,s ( k ) ≤ y v , ∀ v ∈ V , ∀ k ∈ K , ∀ s ∈ F ( k ) . (2)The node capacity constraints can be written as follows: X k ∈K X s ∈F ( k ) λ s ( k ) x v,s ( k ) ≤ µ v y v , ∀ v ∈ V . (3) • Traffic Routing
Let ( k, s ) denote the flow which is routed between the two cloudnodes hosting two adjacent functions f ks and f ks +1 . Similar to [2], wesuppose that there are at most P paths that can be used to route flow ( k, s ) and denote P = { , . . . , P } . Let r ( k, s, p ) be the fractionof data rate λ ks on the p -th path of flow ( k, s ) . Then, the followingconstraint enforces that the total data rate between the two nodeshosting functions f ks and f ks +1 is equal to λ s ( k ) : X p ∈P r ( k, s, p ) = 1 , ∀ k ∈ K , ∀ s ∈ F ( k ) ∪ { } . (4)Let z ij ( k, s, p ) ∈ { , } denote whether or not link ( i, j ) is on the p -th path of flow ( k, s ) and r ij ( k, s, p ) be the associated fraction ofdata rate λ ks . Then r ij ( k, s, p ) = r ( k, s, p ) z ij ( k, s, p ) , ∀ ( i, j ) ∈ L , ∀ k ∈ K , ∀ s ∈ F ( k ) ∪ { } , ∀ p ∈ P . (5)The total data rates on link ( i, j ) is upper bounded by capacity C ij : X k ∈K X s ∈F ( k ) ∪{ } X p ∈P λ s ( k ) r ij ( k, s, p ) ≤ C ij , ∀ ( i, j ) ∈ L . (6)To ensure that the functions of each flow k are processed in theprespecified order f k → f k → · · · → f kℓ k and for each s ∈ F ( k ) ∪{ } and p ∈ P , { ( i, j ) : z ij ( k, s, p ) = 1 } forms a path, we needthe flow conservation constraint (7). Let θ ( k, s ) denote the communication delay due to the trafficflow from the cloud node hosting function f ks to the cloud node host-ing function f ks +1 . Then θ ( k, s ) ≥ X ( i,j ) ∈L d ij z ij ( k, s, p ) , ∀ k ∈ K , ∀ s ∈ F ( k ) ∪ { } , ∀ p ∈ P . (8)To ensure that flow k ’s E2E delay is less than or equal to its threshold Θ k , we need the following constraint: θ N ( k ) + θ L ( k ) ≤ Θ k , ∀ k ∈ K , (9)where θ N ( k ) = P v ∈V P s ∈F ( k ) d v,s ( k ) x v,s ( k ) and θ L ( k ) = P s ∈F ( k ) ∪{ } θ ( k, s ) are the total NFV delay on the nodes and thetotal communication delay on the links of flow k , respectively. • Problem Formulation
The network slicing problem is to minimize a weighted sum ofthe total power consumption of the whole cloud network and thetotal delay of all services: min x , y , r , z , θ X v ∈V y v + σ X k ∈K ( θ L ( k ) + θ N ( k )) s.t. (1) − (9) ,x v,s ( k ) , y v ∈ { , } , ∀ k ∈ K , s ∈ F ( k ) , v ∈ V ,r ( k, s, p ) , r ij ( k, s, p ) ≥ , z ij ( k, s, p ) ∈ { , } , ∀ ( i, j ) ∈ L , k ∈ K , s ∈ F ( k ) ∪ { } , p ∈ P ,θ ( k, s ) ≥ , ∀ k ∈ K , s ∈ F ( k ) ∪ { } , (NS)where σ is a constant value that balances the two terms in the ob-jective function. It has been shown in [2] that problem (NS) canbe equivalently reformulated as an MILP problem and thus can besolved using standard MILP solvers like Gurobi.The following Theorem 1 shows the (strong) NP-hardness ofproblem (NS) in two very special cases and thus reveals the intrinsicdifficulty of solving it. This motivates us to develop efficient algo-rithms for approximately solving problem (NS), especially when theproblem’s dimension is large. Theorem 1. (i) Problem (NS) is NP-hard even when there is only asingle service. (ii) Problem (NS) is strongly NP-hard even when eachnode’s capacity, link’s capacity, and service’s E2E delay thresholdare infinity. Moreover, there does not exist a constant approximationalgorithm to solve it in this case.
3. AN LP ROUNDING-AND-REFINEMENT ALGORITHM
In this section, we focus on designing an efficient algorithm to obtaina high-quality solution for problem (NS). To do this, we first derive acompact LP relaxation for the problem and then develop a two-stageLP rounding-and-refinement algorithm based on it. The basic idea ofthe proposed algorithm is to decompose the hard problem (NS) intotwo relatively easy subproblems and solve two subproblems sepa-rately while taking their connection into account . Specifically, inthe first stage, we find a binary vector (¯ x , ¯ y ) for the VNF placementsubproblem (i.e., (¯ x , ¯ y ) satisfying constraints (1)-(3)) using an itera-tive LP rounding procedure, which takes traffic routing into account.In the second stage, based on the binary vector (¯ x , ¯ y ) , we use anLP refinement procedure to solve the traffic routing subproblem toobtain a solution that satisfies the E2E delay constraints (8)-(9) of allservices. • A Compact LP Relaxation j :( j,i ) ∈L z ji ( k, s, p ) − X j :( i,j ) ∈L z ij ( k, s, p ) = (cid:26) , if i ∈ I\V ; x i,s +1 ( k ) − x i,s ( k ) , if i ∈ V , ∀ k ∈ K , ∀ s ∈ F ( k ) ∪ { } , ∀ p ∈ P . (7)As problem (NS) can be reformulated as an MILP problem[2], simply relaxing the binary variables { y v } , { x v,s ( k ) } , and { z ij ( k, s, p ) } to be continuous variables will give a natural LP re-laxation. Recall that in problem (NS), in order to model differentpaths for flow ( k, s ) , we introduce the notation { p : p ∈ P} and use { ( i, j ) : r ij ( k, s, p ) > } to represent the p -th path of flow ( k, s ) (cf. (5) and (7)). However, as z ij ( k, s, p ) ∈ [0 , in the above nat-ural LP relaxation, the traffic flow { ( i, j ) : r ij ( k, s, p ) > } can besplit into multiple paths. This reveals that there is some redundancyin the natural LP relaxation, i.e., we do not need to introduce thenotation { p : p ∈ P} to model different paths for flow ( k, s ) in it.Inspired by this observation, below we derive a compact LP re-laxation for problem (NS). Our strategy is to simply set P = { } .Then by (4), we have r ( k, s,
1) = 1 , and hence constraint (5) re-duces to r ij ( k, s,
1) = z ij ( k, s, . Furthermore, we can removeconstraints (4), (5), and variables { r ij ( k, s, p ) } , and replace con-straint (6) by X k ∈K X s ∈F ( k ) ∪{ } λ s ( k ) z ij ( k, s, ≤ C ij , ∀ ( i, j ) ∈ L . (6’)The natural LP relaxation then reduces to min x , y , z , θ X v ∈V y v + σ X k ∈K ( θ L ( k ) + θ N ( k )) s.t. (1) − (3) , (6’) , (7) − (9) ,x v,s ( k ) , y v ∈ [0 , , ∀ v ∈ V , k ∈ K , s ∈ F ( k ) ,z ij ( k, s, ∈ [0 , , θ ( k, s ) ≥ , ∀ ( i, j ) ∈ L , k ∈ K , s ∈ F ( k ) ∪ { } . (NS-LP) Theorem 2.
The LP problem (NS-LP) is a relaxation of problem (NS) with any P ≥ . Theorem 2 shows that the above LP problem (NS-LP) is also arelaxation of problem (NS). Note that the numbers of variables andconstraints in problem (NS-LP) are much smaller than those in thenatural LP relaxation of problem (NS), especially when P is large.As a result, solving problem (NS-LP) should be much more efficientthan solving the natural LP relaxation. • Solving the VNF Placement Subproblem
Next, we solve the VNF placement subproblem by constructinga binary vector (¯ x , ¯ y ) that satisfies constraints (1)-(3). Since vec-tor ¯ y can be uniquely determined by vector ¯ x , in the following weconcentrate on constructing the binary vector ¯ x . To do this, we firstsolve the LP relaxation problem (NS-LP), denoted its solution by ( x ∗ , y ∗ , z ∗ , θ ∗ ) . If x ∗ is a binary vector, we obtain a feasible solu-tion ¯ x := x ∗ for the VNF placement subproblem. Otherwise, we set x v,s ( k ) = 1 in problem (NS-LP) if x ∗ v,s ( k ) = 1 . Then we chooseone variable, denoted as x v ,s ( k ) , whose value x ∗ v ,s ( k ) is thelargest among the remaining variables, i.e., x ∗ v ,s ( k ) = max (cid:8) x ∗ v,s ( k ) : 0 < x ∗ v,s ( k ) < , v ∈ V ,k ∈ K , s ∈ F ( k ) (cid:9) . (10)Next we decide to round variable x v ,s ( k ) to one or zero. In par-ticular, we first set x v ,s ( k ) = 1 in problem (NS-LP). If the mod-ified LP is infeasible, we set x v ,s ( k ) = 0 and continue to roundother variables respect to the values { x ∗ v,s ( k ) } . Otherwise, the mod-ified LP is feasible and we repeat the above procedure to the solutionof the modified LP until a binary solution is obtained. The details are summarized in the following Algorithm 1. Algorithm 1
An iterative LP rounding procedure for solving theVNF placement subproblem Initialize the set A = ∅ ; Solve problem (NS-LP) to obtain its solution ( x ∗ , y ∗ , z ∗ , θ ∗ ) ; while (there exists some v ∈ V , k ∈ K , and s ∈ F ( k ) such that < x ∗ v,s ( k ) < ) do For each v ∈ V , k ∈ K , and s ∈ F ( k ) with x ∗ v,s ( k ) = 1 , ifconstraint x v,s ( k ) = 1 is not in set A , add it into set A ; Let ( v , s , k ) be the tuple in (10). Add constraint x v ,s ( k ) = 1 into set A ; Add the constraints in set A into problem (NS-LP) to obtaina modified LP; If the modifed LP is feasible, let ( x ∗ , y ∗ , r ∗ , θ ∗ ) be its so-lution; otherwise, replace constraint x v ,s ( k ) = 1 by con-straint x v ,s ( k ) = 0 in set A and set x ∗ v ,s ( k ) ← ; end while If vector ( x ∗ , y ∗ ) satisfies constraints (1)-(3), then the binaryvector (¯ x , ¯ y ) ← ( x ∗ , y ∗ ) is feasible for the VNF placementsubproblem; otherwise declare that the algorithm fails to find afeasible solution.The above rounding strategy makes sure that we can round onevariable, taking a fractional value at the current solution, at a timeand more importantly this variable can be rounded to a binary valuethat is consistent to other already rounded variables. This is in sharpcontrast to the algorithm in [12] where the variables are roundedwithout ensuring the consistency of the current rounding variablewith other already rounded variables. It is worth remarking that ourrounding strategy takes traffic routing into account (as the modifiedLP contains the information of traffic routing of all services). • Solving the Traffic Routing Subproblem
Once we get a binary vector (¯ x , ¯ y ) , we still need to solve thetraffic routing subproblem by fixing x = ¯ x and y = ¯ y in prob-lem (NS). In this case, the objective function in problem (NS) re-duces to g ( θ ) = P k ∈K θ L ( k ) . Similarly, we solve the LP problem(NS-LP) with x = ¯ x and y = ¯ y to obtain a solution ( z ∗ , θ ∗ ) .Due to the (possible) fractional values of { z ∗ ij ( k, s, } , θ ∗ ( k, s ) can be larger than the communication delay incurred by the trafficflow from the node hosting function f ks to the node hosting function f ks +1 . To recompute the communication delay based on solution ( z ∗ , θ ∗ ) , we need to solve the NP-hard Min-Max-Delay problem[24]. Fortunately, there exists an efficient polynomial-time ( ǫ )-approximation algorithm for this problem [24]. After recomputingthe communication delays between all pairs of nodes hosting twoadjacent functions, we can compute the total delay of each service k , denoted as ¯ θ ( k ) . If ¯ θ ( k ) > Θ( k ) for some service k , the currentrouting strategy is infeasible as it violates the E2E delay constraint ofservice k . We then use an iterative LP refinement procedure to try toget a solution that satisfies the E2E delay constraints of all services.The idea of our refinement procedure is to increase the weightsof the variables θ L ( k ) corresponding to the service whose E2E de-lay constraint is not satisfied at the current solution, in order to refinethe solution. In particular, we change the objective function g ( θ ) inproblem (NS-LP) into ˆ g ( θ ) = P k ∈K ω k θ L ( k ) where ω k ≥ forall k ∈ K . At each iteration, we solve problem (NS-LP) (with theobjective function ˆ g ( θ ) , x = ¯ x , and y = ¯ y ) to obtain its solution Number of Services N u m be r o f F ea s i b l e P r ob l e m I n s t an c e s LPRRLPREXACT (a)
Number of Services N u m be r o f A c t i v a t ed C l oud N ode s LPRRLPREXACT (b)
Number of Services C P U T i m e R a t i o LPRRLPREXACT (c)
Fig. 1 : Performance comparison of LPRR, LPR, and EXACT: (a) Number of feasible problem instances; (b) Average number of activatedcloud nodes; (c) Average CPU time ratio. ( z ∗ , θ ∗ ) . If, for some service k , the E2E delay constraint is vio-lated at this solution, we increase ω k by a factor of ρ > , and solveproblem (NS-LP) again. The procedure is repeated until the solu-tion satisfies the E2E delay constraints of all services or the iterationnumber reaches a predefined parameter IterMax. We summarize theabove procedure in Algorithm 2. Algorithm 2
An iterative LP refinement procedure for solving thetraffic routing subproblem Set ω k = 1 for all k ∈ K , ρ > , IterMax ≥ , and t = 0 ; while t < IterMax do Solve problem (NS) (with the objective function ˆ g ( θ ) , x =¯ x , and y = ¯ y ) to obtain its solution ( z ∗ , θ ∗ ) ; For each service k ∈ K , compute the total delay ¯ θ ( k ) basedon ( z ∗ , θ ∗ ) and (¯ x , ¯ y ) . If ¯ θ ( k ) ≤ Θ k for all k ∈ K , we stopwith the feasible solution ( z ∗ , θ ∗ ) ; otherwise, for each k ∈ K with ¯ θ ( k ) > Θ k , we update ω k ← ρω k . Set t ← t + 1 . end while • Complexity Analysis
The dominant computational cost of our algorithm is to solve theLP problems in form of (NS-LP). The number of solving problems(NS-LP) in Algorithms 1 and 2 are upper bounded by |V| P k ∈K ℓ k and IterMax, respectively. Since an LP can be solved using the(polynomial-time) interior-point method [25], it follows that theworst-case complexity of our proposed algorithm is polynomial.In sharp contrast, the worst-case complexity of using the standardMILP solvers like Gurobi [19] to solve problem (NS) is exponential.
4. NUMERICAL SIMULATION
In this section, we present simulation results to illustrate the effec-tiveness and efficiency of our proposed LP rounding-and-refinement(LPRR) algorithm for solving the network slicing problem. We com-pare our proposed algorithm with the LP rounding (LPR) algorithmin [12] and the exact approach using standard MILP solvers (calledEXACT) in [2]. We choose σ = 0 . and |P| = 2 in problem(NS). In Algorithm 2, we set ρ = 2 and IterMax = 5 . We useGurobi 9.0.1 [19] to solve all MILP and LP problems. When solvingthe MILP problems, we set a time limit of 1800 seconds for Gurobi.We test all algorithms on the fish network topology [11], whichcontains 112 nodes and 440 links, including 6 cloud nodes. Thecloud nodes’ and links’ capacities are randomly generated within [50 , and [5 , , respectively. The NFV and communicationdelays on the cloud nodes and links are randomly generated within { , , , } and { , } , respectively. For each service k , node S ( k ) is randomly chosen from the available nodes and node D ( k ) is setto be the common destination node; SFC F ( k ) is a sequence offunctions randomly generated from { f , . . . , f } with |F ( k ) | = 3 ; λ s ( k ) ’s are the service function rates which are all set to be thesame integer value, randomly generated within [1 , ; Θ k is set to
20 + (3 ∗ dist k + α ) where dist k is the delay of the shortest pathbetween nodes S ( k ) and D ( k ) and α is randomly chosen in [0 , .The above parameters are carefully chosen to make sure that the con-straints in problem (NS) are neither too tight nor too loose. For eachfixed number of services, 100 problem instances are randomly gener-ated and the results presented below are obtained by averaging overthese problem instances.Fig. 1 plots the performance of LPRR, LPR, and EXACT.We can clearly see the effectiveness of our proposed algorithmLPRR over LPR in Figs. 1(a) and 1(b). In particular, as shown inFig. 1(a), using our proposed algorithm LPRR, we can find feasiblesolutions for much more problem instances, compared with usingLPR. Indeed, LPRR finds feasible solutions for almost all feasibleproblem instances (as EXACT is able to find feasible solutions forall feasible problem instances and the difference of the number offeasible problem instances solved by EXACT and LPRR is small inFig. 1(a)). In addition, using LPRR, the number of activated cloudnodes is much smaller than that of using LRP, as shown in Fig. 1(b).The comparison of the solution efficiency of LPRR, LPR, andEXACT is plot in Fig. 1(c). Here we scale the solution time of LPRto be 1 and compute the CPU time ratio as follows:CPU time ratio = T(LPRR) (or T(EXACT))T(LPR) , where T(LPR), T(LPRR), and T(EXACT) are the CPU time takenby LPR, LPRR, and EXACT, respectively. Fig. 1(c) shows that ourproposed algorithm LPRR is much more computationally efficientthan EXACT, and the solution efficiency of LPRR and LPR is com-parable. Indeed, LPRR is at most four times slower than LPR in allcases while EXACT is even 100+ times slower than LPR when theproblem is large (i.e., |K| ≥ ).In summary, our simulation results illustrate the effectivenessand efficiency of our proposed algorithm LPRR. More specifically,compared with LPR in [12], it is able to find a much better solution;compared with EXACT in [2], it is much more computationally ef-ficient. . REFERENCES [1] R. Mijumbi, J. Serrat, J.-L. Gorricho, N. Bouten, F. De Turck,and R. Boutaba, “Network function virtualization: State-of-the-art and research challenges,” IEEE Communications Surveys &Tutorials , vol. 18, no. 1, pp. 236-262, Firstquarter 2016.[2] W.-K. Chen, Y.-F. Liu, A. De Domenico, Z.-Q. Luo, and Y.-H. Dai. “Optimal network slicing for service-oriented networkswith flexible routing and guaranteed E2E latency,” 2020. [On-line]. Available: https://arxiv.org/abs/2006.13019.[3] W.-K. Chen, Y.-F. Liu, A. De Domenico, and Z.-Q. Luo, “Net-work slicing for service-oriented networks with flexible routingand guaranteed E2E latency,” in
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