networks volume 31 issue 4 1998 [doi...

A New Proximal Decomposition Algorithm forRouting in Telecommunication Networks

P. Mahey,1 A. Ouorou,1 L. LeBlanc,2 J. Chifflet3

1 Laboratoire inter-universitaire dInformatique, de Modelisation et dOptimisation desSyste`mes, Universite Blaise Pascal, ISIMA, BP 125, 63173 Aubie`re cedex, France

2 Owen Graduate School of Management, Vanderbilt University, Nashville, Tennessee 37203

3 CNET, 305 rue A. Einstein, 06927 Sophia-Antipolis, France

Received 6 May 1996; accepted 16 December 1997

Abstract: We present a new and much more efficient implementation of the proximal decompositionalgorithm for routing in congested telecommunication networks. The routing model that we analyze is astatic one intended for use as a subproblem in a network design context. After describing our newimplementation of the proximal decomposition algorithm and reviewing the flow deviation algorithm, wecompare the solution times for (1) the original proximal decomposition algorithm, (2) our new implementa-tion of the proximal decomposition algorithm, and (3) the flow deviation algorithm. We report extensivecomputational comparisons of solution times using actual and randomly generated networks. Theseresults show that our new proximal decomposition algorithm is substantially faster than the earlier proximaldecomposition algorithm in every case. Our new proximal decomposition is also faster than the flowdeviation algorithm if the network is not too congested and a highly accurate solution is desired, suchas one within 0.1% of optimality. For moderate accuracy requirements, such as 1.0% optimality, and forcongested networks, the flow deviation algorithm is faster. More importantly, solutions that we obtainedfrom the proximal decomposition algorithm always involve flow on only a small number of routes betweensourcedestination pairs. The flow deviation algorithm, however, can produce solutions with flows on avery large number of different routes between individual sourcedestination pairs. q 1998 John Wiley &Sons, Inc. Networks 31: 227238, 1998

1. INTRODUCTION point data requirements. Although this algorithm was ef-ficient for solving these problems, its computational re-

In [4] , a proximal decomposition algorithm was proposed quirements were greater than those of the flow deviationto solve convex multicommodity flow problems of min- algorithm (see [8] or [18]) on real-size networks. Inimizing average packet delay in networks with point-to- this paper, we propose a new and much more efficient

implementation of the proximal decomposition algorithmfor these multicommodity flow problems. This implemen-

Correspondence to: P. Mahey; e-mail: Philippe, [email protected] tation handles larger networks with a larger number ofContractgrantsponsor:CentreNationaldEtudesdesTelecommunica-commodities ( i.e., required communications betweentions

Contract grant number: 935B. nodes) , even with required communication between each

q 1998 John Wiley & Sons, Inc. CCC 0028-3045/98/040227-12

227

8U22 0816/ 8U22$$0816 05-14-98 16:24:35 netwa W: Networks

228 MAHEY ET AL.

pair of nodes and congested traffic, in competitive CPU do not consider here node capacities) carrying a flow fis proportional totimes. These results are quite important for finding opti-

mal routing in a static model with relatively high accuracyrequirements such as data communication networks.

F( f ) fc 0 f .In section 2, we give equivalent alternative formula-

tions of this routing problem with point-to-point data re-quirements and queuing delays. In Sections 3 and 4, we We let G (X , U) represent the directed graph withreview the existing literature and describe our new proxi- capacities cu on each arc uU , and we denote the numbermal decomposition algorithm and the flow deviation algo- of arcs by n . For k 1, . . . , K , let f ku be the quantityrithm. In Section 5, we compare the numerical behavior of commodity k which flows through arc u , that is, theof these methods with respect to CPU times and sensitiv- quantity of packets associated with demand k whose routeity to load and sparsity of the demand. We report exten- includes link u , and let f k R n be the vector of thesesive computational results on randomly generated net- flow values for each k . Finally, let fu be the total (aggre-works with up to 500 nodes and on real networks with gate) flow flowing through arc u . We have seen that theup to 106 nodes and a fully dense requirement matrix. delay caused by the saturation of traffic on arc u is Fu( fu)

fu / (cu 0 fu) . Then, the problem of minimizing theglobal delay incurred by packets sent through the networkcan be modeled by the following multicommodity flow2. THE ROUTING MODELproblem:

In our routing model, packets can take any number of (MCF) :different routes between each source and destination; thisis referred to as multipath routing or bifurca-

min n

u1Fu( fu)tion. In a datagram network, packets between a specific

source and destination can literally take different routessimultaneously because of congestion. Thus, the multi-

s.t. f k Vk k 1, . . . , K (1)path assumption inherent in our routing model is appro-priate when modeling these systems.

Packet-switched virtual circuit networks are more K

k1f ku fu u U (2)

common than are datagram systems, and in the formernetworks, packets between each user and the destinationtake a single route. Nevertheless, the bifurcation assump- 0 fu cu u U , (3)tion is usually appropriate even when modeling these net-works, since each node typically represents a LAN or at where Vk is the set of flow vectors for demand k satisfyingleast a building with multiple-user terminals. When one conservation of flows:user accesses a remote node, he or she is given one spe-cific route. Later, while this user is still communicating Vk { f k R nM f k Dk , 0 f k}.with the destination node, another user at the same originaccessing the same destination may be given another route M is the incidence matrix of the graph,because of congestion. Thus, different routes are simulta-neously used in practice between the same origin node Dk (0, . . . , 0dk ,

source node. . . , dk ,

demand node. . . , 0) ,

and destination node. For these reasons, our continuousstatic routing model with multipath routing is again appro-priate. and dk is the required amount of flow between the k th

source and destination pair (referred to as an SD-pairThe network is represented by a graph with capacitatedarcs. To model the required traffic, we associate with each hereafter) .

The model above is the one which was used in [4] topair of nodes ( i , j) a required flow value dk (possibly 0).We use the convention that the k th commodity denotes derive a proximal decomposition algorithm based on the

relaxation of the coupling subspace constraints (2) . Thethe amount of traffic (in bits /second) flowing from nodei to node j . In a packet switched network, some applica- new implementation of the proximal decomposition algo-

rithm that we are proposing here is based on an arctions (e.g., compressed video images) are very sensitiveto the loss of packets occurring when a buffer at some path formulation of the multicommodity flow model

(MCF). This alternative model, which we now explain,node saturates. To avoid these situations, we minimizethe average delay on each arc as suggested by Kleinrock is entirely equivalent to (MCF).

The flow variables are now indexed by k and p , where[16]. The average delay on a single arc of capacity c (we


ROUTING IN TELECOMMUNICATION NETWORKS 229

p is the index associated with the p th path between the straints and by the use of dual ascent directions or ofk th SD-pair. Let xkp represent the portion of the k th com- contractive maps to reach equilibrium.munication flow between the k th SD pair routed on path In the first class of algorithms, the most commonlyp . An implicit formulation of the problem (MCF) is to used method is the flow deviation algorithm. Thisuse all possible paths between each SD-pair by defining is a network version of the FrankWolfe method forthe coefficients pkp(u) of the arcpath incidence matrix linearly constrained optimization problems (see [18] andas [8] for a typical adaptation to the routing models above).

Many variants and accelerated versions of the flow devia-tion method have been suggested in the literature, includ-

pkp(u) H1 if path p for the k th SD-pair uses arc u0 else. ing the parallel tangents method [7, 19]. An excellentsurvey of routing algorithms for transportation networkswas given by Florian and Hearn [6] .

Other efficient descent procedures for solving the rout-A specific path is then associated with both indexes k anding problem are the distributed projected-gradient methodp and will be denoted by pathkp . pkp denotes the numberof Gallager [12] and the projected Newton method de-of arcs of pathkp .signed by Bertsekas and Gafni [3] . A recent attempt toIn our implementation, we do not enumerate paths asolve multicommodity flow problems by known optimiza-priori; instead, we iteratively generate paths when

solving the subproblems induced by the proximal decom- tion techniques is the interior point approach by Schultzposition. To get a complete description of the implicit and Meyer [25].formulation, we still need the aggregate flow variables fu , In the second class of algorithms, some of the con-which will be treated separately on each arc after decom- straints are relaxed to yield a concave dual problem toposition: which efficient optimization techniques may be applied.

In some cases, Lagrangian relaxation is used to decouple(APMCF) ArcPath Model for (MCF): the individual commodities by duality. Examples are the

restricted simplicial decomposition (see [14] and [17]) ,which is an adaptation of the DantzigWolfe decomposi-min

n

u1Fu( fu)

tion principle to convex costs, and the subgradient ap-proach in [9] . Besides these classical dual schemes, manyalgorithms involve the network relaxation method (see

s.t. K

k1

pNkpkp(u)xkp fu u U (4) [2]) which is a coordinate or block-coordinate descent

technique applied in general to the dual of (MCF) wherethe computations affect the nodes or groups of nodes

pNkxkp dk k

residuals ( i.e., constraint violations) . The efficiency ofdual schemes depends highly on the smoothness of the

0 fu cu u U (5) dual function, which is true with a strictly convex objec-tive function. Observe that the objective function Fu in0 xkp k , p ,model (MCF) is not strictly convex with respect to allvariables since it does not depend on the flow variableswhere Nk is the set of paths between the k th SD-pair.f ku . Nagamochi [21] introduced strictly convex costs onthe individual flows in the objective function to get asmooth dual (he also added too some capacity bounds on3. LITERATURE REVIEWeach individual flow to get a more general formulation).Dual ascent techniques for network optimization (differ-The routing models presented in the previous section haveent from the previous ones since duality here concernsa significant internal structure which has been used in theall the flow equations, not just the multicommodity flowpast to build various algorithms. The structural aspectsconstraints) were implemented early by Stern [27] onthat can be exploited are decomposition into individualtelecommunications networks and later by Tseng and Ber-flow problems, monotropic programming (i.e., separabil-tsekas [29]. A general framework for dual ascent meth-ity of the objective function with respect to the variablesods was proposed by Tseng [28]. Similar to these tech-and linear constraints) , and path generation by minimal-niques is the row-action method developed for nonlinearcost path computation. We may distinguish two classestransportation models by Zenios and Censor [30]. Asof algorithms: Class 1 consists of direct methods whichshown by these last authors, most of these algorithmstry to adapt known mathematical programming tech-present very favorable features for massively parallelniques to the network structure. Class 2 consists of dual

methods which differ by the choice of the relaxed con- computing. Indeed, the flow computations are distributed


230 MAHEY ET AL.

arcwise like in iterative relaxation methods for systems 1. Subproblem: For the feasible solution x t , solveof linear equations.

The last class of methods which includes the proximal Minz F(x t) / F(x t)(z 0 x t) subject todecomposition algorithm described below is also amena-

Az b , z 0,ble to distributed computations. But, as they are exten-sions of the proximal point algorithm for constrained con-vex programs, they are well adapted to nonlinear network or omitting the constant terms,models (see [1]) . Initially, Spingarn [26] proposed thepartial inverse method for separable convex program- Minz F(x t)z subject to Az b , z 0.ming. Later, Eckstein showed the relation it bears withthe alternating direction of multipliers method (see [11],

Let z t be the optimal subproblem solution.[5] , or [10]) . This is the approach that we have chosen2. Line search: Define the search direction to be z tin [4] to solve the (MCF) and we will extend it here to

0 x t and perform a line search, getting the best stepthe arcpath model (APMCF).size and new solution:

4. THE TWO BASIC ALGORITHMS at arg mina [0,1]

F(x t / a(z t 0 x t))We present here briefly two of the algorithms which we x t/1 x t / at(z t 0 x t) .have compared for solving the multicommodity flowproblem (MCF): the flow deviation method (FD) and 3. Lower bound: Compute the lower bound F(x t)our new implementation of the proximal decomposition / F(x t)(z t 0 x t) .method (PDM). The reader is referred to [4] for the

4. Convergence check: Stop if F(x t) is within eoriginal version of (PDM).of the largest lower bound found at any iteration;otherwise, set t : t / 1 and go to 1.

4.1. The Flow-deviation Algorithm (FD)

This algorithm treats the link capacities (3) indirectly. The subproblem is solved by finding the minimal-costSince we begin with a feasible solution, the capacity will route between each SD-pair and sending the requirednever be exceeded at subsequent iterations because the packets along these routes.cost tends to infinity when we get close to that capacity. The lower bound in step 3 is valid since the optimalThe flow deviation algorithm (Fratta et al. [8]; LeBlanc solution x* satisfies[18]) iteratively solves

F(x*) F(x t) / F(x t)(x* 0 x t) by convexity of F1. A direction-finding subproblem.

F(x t) / F(x t)(z t 0 x t) by definition of z t .2. A line-search in the resulting direction.

An alternative is to utilize Wardrops equilibrium condi-The subproblem is solved by finding the minimal-costtions directly by measuring the difference in the costs ofroute between each SD-pair and loading the requiredthe routes used within an SD-pair [23].amount of flow onto the corresponding routes. The link

In our model, we store and update the individual pathcosts used when finding the cheapest routes are the partialflows as well as the aggregate flow in order to count thederivatives of the objective function Fu( fu) evaluated atnumber of paths used. Observe that, since x t is a feasiblethe current solution. More specifically, the algorithmflow and z t is an extremal feasible solution (one shortestworks on a general linearly constrained problemroute for each commodity) , the line search on [x t , z t] willkeep all previously generated path flows strictly positive,Min F(x) subject to Ax b , x 0,unless at 1, which is unlikely in practical situations.This point is important for understanding why the flowwhere F is continuously differentiable. (FD) successivelydeviation method generates a high number of active pathssolves linearized subproblems to get a feasible descent(see Section 5).direction; the steps in this algorithm are

This algorithm is very efficient for finding near-optimalsolutions, such as within 1 or 2% of optimality. It is used(FD)

0. Initialization: Set the iteration index t 0, to solve a routing model for a 30,000-link, 12,000-nodetransportation model of the streets of Chicago on a regularfind a feasible initial solution x 0 , and choose the con-

vergence parameter e. basis.



4.2. The Proximal Decomposition Method the subspace A represents the coupling between each sub-system. Many distinct strategies are possible to put the(PDM)multicommodity flow problem (MCF) in the form (P).The (PDM) algorithm is a specialized version of the par- In [4] , the subspace represents the coupling betweential inverse method designed by Spingarn [26] for thecommodities [Eqs. (2)] . Eckstein [5] showed that in-decomposition of convex separable problems. It was ini-cluding node balance equations in the subspace can leadtially designed to solve a generic convex constrainedto specific implementations of (PDM) which are equiva-problem: lent to the alternating direction method of multipliers(ADMM) earlier designed for variational inequalities(see [11]) . This is the way that we have now chosen to(P) min F(x)

x A , implement (PDM) on multicommodity flow problems.Moreover, we show in this paper that much smaller CPU

where F is convex lower-semicontinuous and A is a closed times are required when the arcpath formulationsubspace. We denote by A the orthogonal subspace to (APMCF) is used.A . Thus, an optimal primal-dual pair (x , y) must lie in The derivation of our new algorithm stems from thethe Cartesian product space A 1 A . Note that x and application of (PDM) to (APMCF) when the couplingy have the same dimension and must lie in orthogonal subspace includes the Eqs. (4) and (5), not only Eq. (2)subspaces (y is a subgradient of F at x A when opti- as in the original (PD) algorithm. Thus, the new featuresmality is reached). The algorithm which can be seen as of the present implementation with respect to the algo-a constrained version of the proximal point method (see rithm described in [4] are twofold:[24]) performs two distinct steps at each iteration: a prox-

Path generation: New paths are generated by minimal-imal step on the objective function F ( the purpose of thiscost path calculations when optimality conditions arestep is to regularize the objective function by adding anot met by the current active path flow values and theirquadratic term depending on the previous primal-dual pairdual counterparts in (APMCF).of solutions) and a projection step on the corresponding

Distributed updating for flows and potentials: Since thesubspaces. At iteration t , given a primal-dual pair (x t , y t)flow requirements (5) are dualized, each path flow A 1 A and a positive parameter l, the computationvalue and each aggregate arc flow value are updatedof the new updates (x t/1 , y t/1) consists of the followingseparately.steps:

We skip here the details of the successive transformations(PDM) needed to get a closed form of the path flows updates

(see [22]) . As in [4] , the aggregate flows fu are updatedProximal step: on each arc by solving one-dimensional strongly convex

subproblems. The new feature concerns the individualcommodity flow updates on each existing path: They areut arg min

uHF(u) / 12l \u 0 x t 0 ly t\ 2J now updated explicitly without the need for solving qua-

dratic flow subproblems, which were the most time-con-suming steps in the original proximal decomposition algo-

t 1l

(x t / ly t 0 ut) .rithm. As observed before, we obtain the same kind ofdistributed updating formulas as the ones used in the alter-nate direction of multipliers method described in [5] .Projection step:

We focus now on the iterative generation of the pathsuntil optimality is attained. We use the following notation:(x t/1 , y t/1) ProjA1A (ut , t) .

Let Nk denote the number of paths in Nk , and n , thenumber of arcs.The behavior of this algorithm and its numerical perfor-

The matrix A of coefficients of the coupling constraintsmance have received only limited attention on specifichas n / K rows and n / (Kk1 Nk columns. It defines themodels like the FermatWeber location problem [15],coupling subspace A and has the following structure [seequadratic cost transportation networks [5] , and multi-(4) and (5)]:commodity flow problems [4]. In [20], theoretical results

show how the sensitivity to the scaling parameter l canbe measured in specific situations, basically when F isstrongly convex with a Lipschitzian gradient.

A

I 0p1 0p2 ??? 0pK0 w1 0 ??? 00 0 w2 ??? :: : ??? ??? 00 0 ??? 0 wK

,The key fact lies in the adaptation of complex, specificreal models to the generic problem (P). In most situa-tions, F is a separable function on a product space and


232 MAHEY ET AL.

where pk is the n by Nk matrix whose (u , p) entry is 1. For each arc u ,pkp(u) and wk are Nk-row vectors with coefficients 1.

The primal variables are denoted by f t/1u arg min0fucu

HFu( fu) 0 y tu fux ( f1 , . . . , fn , x11 , . . . , x1N1 , . . . , xK1 , . . . , xKNK ) ,

/ l2 S( fu)2 0 2S f tu / ru(x

t)d(u) D fuDJ .

(8)and the dual variables associated with constraints (4) and(5), by

2. Compute the new arc costs F *u ( f t/1u ) . Then,y (y1 , . . . , yn , Y1 , . . . , YK) . determine the shortest path path t/1k be-

tween each SD-pair using the new arcThe residuals [violation of constraints (4) and (5)] are costs.denoted by If path t/1k / Nk , Nk : Nk < {path t/1k } and in-

crement the number of paths:Nt/1k Ntk / 1.ru(x)

K

k1Nk

p1pkp(u)xkp 0 fu (6) Compute the new path flows

rk(x) dk 0 Nk

p1xkp (7)

x t/1kp x tkp / 1l(1 / pkp) (Y

tk 0

upathkpy tu)

and d(u) 1 / Number of paths using arc u ./ 1

1 / pkp S rk(xt)

Nt/1k0

upathkp

ru(x t)d(u) D pAs Nk , the set of paths for the k th SD-pair is not known

a priori; we substitute it at each iteration t 0, 1, rrr bya subset Nkt Nk which contains the previously generated 1, . . . , Nt/1kpaths. The number of paths in Nkt is denoted by Ntk .

The initialization procedure computes minimal-costpaths between each SD-pair where the costs on the arcs If x t/1kp 0, x t/1kp R 0.are the first derivatives at the origin F *u (0) ( the super-scripts correspond hereafter to iteration number):

3. Update the dual variables

Procedure Initialization0. For each arc u , set priceu F *u (0) y t/1u y tu / ld(u) ru(x

t/1) uZero out the path flows x 0 0.

1. For each SD-pair kY t/1k Y tk / lNt/1k rk(x

t/1) k .Compute the cheapest path pathk1 betweenthe k th SD-pair for the arc costs priceuDefine number of paths: N 0k 1

4. Stopping criterionDefine the flows on path 1: x 0k1 dkUpdate the residuals ru(x t/1) , u , rk(x t/1) ,Increment link flows: f 0u f 0u / x 0k1 , for allklinks u on path k1Compute the dual infeasibilitiesRedefine arc costs:

If f 0u cu , define priceu F *u ( f 0u)else, set priceu Infinite.

2. Define dual prices and residuals: s t/1u HF *u ( f t/1u ) 0 y t/1umin(F *u ( f t/1u ) 0 y t/1u , 0)

if f t/1u 0if f t/1u 0For each arc u , set: y 0u priceu

For each SD-pair k , set Y 0k (upathk1 y 0u (9)For each arc u and each SD-pair k , com-pute: ru(x0) , and rk(x0) , using (6) and (7).

Algorithm (APPDM)0. Choose the scaling parameter l 0 and the s t/1kp HY t/1k 0 (upathkp y t/1u

min(Y t/1k 0 (upathkp y t/1u , 0)if x t/1kp 0if x t/1kp 0.convergence parameter . Begin Initializa-

tion.Set the iteration index t 0 (10)



If Nk contains a finite number of paths and the set of gener-ated paths can only be augmented. Since the maximumnumber of paths is bounded, the procedure stops. Letmax(maxus t/1u , maxk ,ps t/1kp ,NU k be the final set of generated paths. When (11) holds,maxuru(x t/1u ), maxkrk(x t/1)) e the following conditions are satisfied:

Stop and save the current solutionOtherwise set t R t / 1 and go to step 1.

Observations: (1) The computation of the minimal-costpaths in step 2 may be completed by sweeping all possibledestinations for a given origin, avoiding the explosion of

F *u ( f tu) y tu if f tu 0F *u ( f tu) y tu if f tu 0Y tk

upathkpy tu if x tkp 0, p NU k

Y tk upathkp

y tu if x tkp 0, p NU k

ru(x t) 0 u rk(x t) 0 k0 f tu cu u 0 x tkp k , p ,

(12)the computational time when the requirement matrix isdense (this is also true for the flow deviation algorithm).(2) The proximal decomposition method is a primal-dualmethod and feasible solutions are only guaranted at theconvergent point. On the other hand, the convergenceanalysis of the original method (discussed first in [26];see also [19]) assumed that the problem possesses feasi-

which are nothing but the KuhnTucker conditions forble solutions. In other words, the general equivalent for- (APMCF) with Nk replaced by the set of generated pathsmulation is minxA F(x) , where F is convex proper lower NU k . Then, a solution of that problem is obtained. Sincesemicontinuous and A is a subspace supposes that dom(F)the paths that have not been generated are such that x tkp> A x M. 0 by construction (see the initialization procedure) ,The algorithm above generates, thus, a primal-dual se-and since the lengths of these paths are greater than orquence { f t , x t , y t , Y t} which converges toward a fixedequal to the shortest path length Y tk , we can extend thepoint and the following theorem shows that it is indeedabove conditions (12) to include all the paths in Nk , andan optimal solution to (APMCF):the theorem is proved. Observe that, for practical imple-mentations, condition (11) is satisfied within a given tol-Theorem 1. Let { f t , x t , y t , Y t} be a sequence generatederance e .by the algorithm above and suppose that for some t

s tu s tkp ru(x tu) rk(x t) 0. (11)5. COMPUTATIONAL RESULTS: ACOMPARATIVE STUDY OF (APPDM)Then, { f t , x t , y t , Y t} is optimal for (APMCF).AND (FD) METHODSProof. The convergence of the proximal decomposi-

tion algorithm was established in [20]. We need only toWe will present three sets of computational tests:prove that when (11) is satisfied, then an optimal solution

is obtained. We show briefly how the monotonic addition A comparison between (APPDM) and the originalof paths in the model will necessarily lead to an active

(PDM) algorithm on small-size networks.set of paths which satisfies within a given tolerance theoptimality conditions for (APMCF). For each SD-pair k , A comparison between (APPDM) and (FD) on large

TABLE I. Comparison of the new implementation (APPDM) with the original (PDM)

Delay CPU Time (in sec) No. of IterationsNo.

Communications APPDM PDM APPDM PDM APPDM PDM

5 1.23 1.24 0.31 37.7 276 4210 2.22 2.23 0.42 70.8 276 5915 3.74 3.74 0.56 177.9 310 9520 5.45 5.45 1.19 283.4 523 11025 7.40 7.40 0.86 471.5 325 12330 8.99 8.99 1.01 587.3 338 135


234 MAHEY ET AL.

TABLE II. APPDM versus FD

4452 SD-pairs

Scale 1.0 1.5 2.0 2.5 3.0Delay 12.59 19.19 25.99 33.01 40.25CPU time (sec) APPDM 22.08 25.92 31.88 31.13 38.9No. iterations APPDM 4 8 26 22 26Path dispersion for APPDM 2 3 2 2 2CPU time (sec) FD 73.5 83.0 92.8 86.3 88.3No. of iterations FD 46 52 58 54 50

6678 SD-pairs

Scale 1.0 1.5 2.0 2.5 3.0Delay 19.65 30.19 41.23 52.82 65.02CPU time (sec) APPDM 33.74 40.32 38.3 66.94 111.33No. iterations APPDM 6 12 10 40 80Path dispersion for APPDM 3 3 2 3 3CPU time (sec) FD 146.2 198.1 187.7 167.2 159.5No. of iterations FD 91 123 115 103 99

8904 SD-pairs

Scale 1.0 1.5 2.0 2.5 3.0Delay 26.48 40.99 56.46 73.01 90.77CPU time (sec) APPDM 79.35 72.65 69.77 82.71 319.54No. iterations APPDM 36 30 28 44 230Path dispersion for APPDM 3 3 3 3 4CPU time (sec) FD 241.9 284.7 271.9 245.6 255.8No. iterations FD 149 176 166 152 158

11130 SD-pairs

Scale 1.0 1.5 2.0 2.5 3.0Delay 33.50 52.29 72.69 94.98 119.47CPU time (sec) APPDM 76.93 83.88 156.49 161.54 712.04No. iterations APPDM 24 36 84 192 494Path dispersion for APPDM 3 3 3 3 4CPU time (sec) FD 355.8 362.7 359.4 347.6 361.3No. iterations FD 219 223 221 213 222

networks with a dense matrix of point-to-point require- gives the maximum number of paths used by a commod-ments. ity. In Table V, flow/cap corresponds to fu /cu .

Table I shows the improvement of our new implemen- A set of tests for (APPDM) running on randomly gen-tation over the original (PDM) implementation. The testserated problemswere performed on relatively small problems19 nodes,68 arcs, and up to 30 commodities. The new algorithmAll tests were performed on a Sun Sparc10/30 work-improves the CPU times by a factor of 100 to 500!station with 32 Mb RAM and the code was entirely writ-

The improvement over (PDM) is due mainly to theten in C.shortcut induced by the arcpath formulation. No qua-The time units reported in the tables are seconds. Thedratic flow subproblems are needed any more, which re-column (or row) headed Delay corresponds to theduces the CPU time drastically even if more iterationsvalues of F( f ) (see Section 2) computed for the solutions

given by the algorithms while the row Path dispersion are necessary to reach the same accuracy.



TABLE III. Example of path dispersion for (APPDM) its proper optimality test, that is, when the maximal toler-and (FD) ance for both KuhnTucker optimality and the row resid-

uals is lower than 1003 . We stopped (FD) when the delayPath Dispersion

value is lower than or equal to the best delay value ob-tained by (APPDM). In Figure 1, we compare the numer-Scale APPDM FDical convergence of both algorithms. Since (FD) gener-

3.5 4 44 ates feasible flows at each iteration, the monotonic de-4.0 6 49 crease of the delay function was chosen to illustrate its4.5 5 52 convergence. No such monotonic decrease can be shown

with (APPDM) iterations, since like all dual methods,5.0 7 51feasibility is not maintained during the iterations. Thus,5.5 7 55we chose the maximal row residual versus the number of6.0 6 55iterations to illustrate the behavior of (APPDM). Whatshould be clear from these graphs are the slow tail of(FD) iterations when the relative error is below 10%

The comparison between (APPDM) and (FD) is and the premature convergence of (APPDM) if looseshown for much larger networks which the original precision requirements are used.(PDM) algorithm was unable to solve efficiently. We We conclude that (FD) and (APPDM) have comple-compared these algorithms for various networks. Repre- mentary behavior in these models, such that (APPDM)sentative results shown in Table II below come from a is faster when a tight precision is required on the stoppinglarge real telecommunications network with 106 nodes criteria. Note also that the performance of (APPDM)and 904 arcs. Two parameters are used to generate the tends to get worse when the network is highly congested.test problems: The average traffic load on the network As was said above, when scale 3, (FD) seems to con-(up to three times the standard demand, corresponding to verge faster. This behavior was confirmed on even moreparameter Scale in Tables II and III) and the density congested networks (scale up to 6). Another significantof the requirement matrix (from 40 to 100%, i.e., a fully fact obtained from these results is the comparative behav-dense matrix. For the latter case, 106 1 105 11130 ior with respect to the number of commodities, that is,commodities must be simultaneously routed on the net- the density of the demand matrix. We first observe thatwork). both methods are able to handle a large number of com-

Table II shows the number of iterations and CPU times modities (up to 11130 in the fully dense case with afor both (APPDM) and (FD). The results show that the communication requirement between every SD-pair) .computational burden per iteration is roughly of the same This is mainly due to the distribution of the computationmagnitude for both algorithms but (APPDM) needs a among the paths which are iteratively generated bylower number of steps to reach the desired accuracy. Tobe more precise, the comparisons rely heavily on theprecision measure which is used to stop the algorithms. 1. Computing the cheapest paths to deviate the aggregate

flow while maintaining the feasibility of the individualSince the stopping criterion is not unique and differs fromone method to the other, we chose to stop (APPDM) at flows in (FD).

Fig. 1. (a ) Row residual versus number of iterations for APPDM. (b) Relative error versusnumber of iterations for FD.


236 MAHEY ET AL.

using the first derivative arc cost values. Then, individualpath flows are updated separately and many of these flowsmay be forced to zero. This intuitive explanation is hereempirically confirmed by our experimentation.

Finally, (APPDM) was tested on larger randomly gen-erated multicommodity flow problems. The network gen-erator is the same as the one used by Goffin et al. [13].The dimensions are given in Table IV and the CPU timesobtained on the same machine as before are shown inTable V. These additional tests were made to prove theability of (APPDM) to be competitive on very large ran-dom problems (a compact formulation of problem 22would involve 81 106 variables and 21 106 constraints) .The results show that the dimension is far to be the mostdeterminant parameter to explain the computational bur-den measured by the CPU times and numbers of iterations

Fig. 2. Sensitivity to the parameter. on the randomly generated networks. These results alsoshow that (APPDM) can solve most large problems withup to 1004 accuracy, which encourages one to investigate

2. Computing the cheapest paths to adjust the individual a more complete comparison with the most efficient algo-flows with the aggregate flow until optimality is rithms on nonlinear multicommodity flow problems.reached in (APPDM). We observed that a small num-ber of such paths is needed in practice. 6. CONCLUSION

We have shown that the arcpath implementation of theThe number of (APPDM) iterations seems to increase proximal decomposition method on the multicommoditywith the number of commodities. However, we muststress that (APPDM) is very sensitive to the choice of

TABLE IV. Description of randomly generatedthe proximal parameter l. We chose to use the same value problemsof the parameter (l 1) in all the tests, but we can seein Figure 2 that that value could be good or bad depending Problem No. Nodes No. Arcs No. Commoditieson the data. The choice of a constant l also explains why

1 60 280 100the CPU times do not always present monotonicity w.r.t.2 60 280 500scale.3 60 280 2000With an empirical adjustment of the l parameter, the4 100 300 120comparison between (APPDM) and (FD) seems to be5 100 600 200favorable to our implementation of the proximal decom-

position when a large number of commodities are to be 6 100 600 200routed simultaneously on the network. Congestion tends 7 200 800 200to give the advantage back to (FD) but we may argue 8 200 800 500that the solution given by (FD) presents much higher 9 200 1000 500dispersion on all possible individual optimal paths. 10 200 1000 1000

Indeed, as we pointed out in Section 4, (FD) tends to 11 200 1000 3000keep positive flows on all generated paths. On the other12 300 1200 1000hand, the optimal solution of the problem (APMCF) is13 300 1600 1000not unique w.r.t. individual path flow values although it14 400 1600 2000is unique w.r.t. aggregate flow values. The final solution15 300 2000 4000given by (FD) is frequently split among alternative paths16 300 2000 1000(see path dispersion in Table III obtained on large net-17 400 2000 2000works with a full dense matrix in the most congested

situations) . 18 400 2000 3000The reason why (APPDM) tends to push flow on a 19 400 2000 4000

fewer number of paths, which means putting as much 20 400 2000 5000flow on as possible arc-disjoint routes, can be explained 21 500 2000 3000by the path generation procedure: Aggregate flow is up- 22 500 2000 4000dated first by (8) and then a new shortest path is computed



TABLE V. APPDM on randomly generated problems

No. CPU Time Path Max [flow/cap] Average [flow/cap]Problems Iterations (in sec) Dispersion Ratio Ratio Delay

1 1708 18.86 3 54.6 26.8 63.152 6975 212.06 2 62.5 31.3 84.603 24,386 2393.19 2 57.1 32.4 97.664 1850 31.41 2 60.3 27.3 85.405 3694 88.87 3 59.9 25.5 100.016 3784 123.05 9 57.7 27.8 104.947 3001 253.66 3 58.5 26.1 163.758 11,278 1095.48 4 60.3 28.9 213.109 22,219 1333.47 4 60.9 28.8 218.74

10 4794 530.49 3 58.5 32.7 246.8911 6437 1366.3 2 62.6 36.5 283.9912 29,192 2918.06 3 62.8 31.4 356.0413 36,471 4907.17 4 63.3 30.0 351.7514 24,173 4692.39 2 65.7 33.5 506.2715 37,380 11,093.2 2 36.1 60.8 430.9716 23,876 5972.72 6 64.7 28.5 359.3917 32,852 8976.65 4 61.0 32.1 515.8618 32,929 8670.13 2 34.8 64.1 533.3619 43,896 14,235.7 2 58.9 35.2 563.7420 34,685 13,628.5 2 60.5 36.6 565.0921 50,00022 44,022 16,952.8 2 62.0 35.0 654.06

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