6
Lagrangian Relaxation in the Multicriterial Routing Krzysztof Stachowiak, Joanna Weissenberg and Piotr Zwierzykowski Pozna´ n University of Technology, Chair of Communication and Computer Networks ul. Polanka 3, 60-965 Pozna´ n, Poland, e-mail: [email protected] Abstract—Communications networks routing is not a new subject. However, with a growth of packet networks and the increase of their range and importance, the task of conveying information over the web has grown beyond its original shape and has become far more sophisticated. It is no longer computer scientists and the military that make use of the global network exclusively; it can be stated that almost every possible group from the global population has numerous representatives among Internet users. This fact is coupled with a decrease in the popularity of the original communication media such as the telephone and the e-mail. This is mainly due to them being assimilated into packet networks along with the process of network convergence. A number of different types of information streams in packet networks has been observed and classified. It turns out that careful resources management itself may lead to an increase in the information transporting efficiency. In this paper, an approach is explored - the Lagrangian relaxation - that enables incorporating multiple criteria into the routing problem solution. At the same time, the low computation complexity is maintained that enables further consideration of the technique also in practical scenarios. Index Terms—routing, Lagrangian relaxation, algorithm I. I NTRODUCTION Communications networks routing is a well known problem that has been given a lot of attention and some well-developed solutions have been provided for solving it. Originally, the routing problem has been modeled as the shortest path finding graph problem which has a good and efficient solution in the Dijkstra’s algorithm, which is broadly used [1], [2]. Its main merit, the simplicity, unfortunately also draws its limits. With communication networks being utilized for more and more purposes, a differentiation between the so-called classes of traffic emerges. Whereas it is still desired to look for the shortest (optimal) path in the network, it is not always obvious what optimization criteria are the most important in a given case, i.e. what does the “shortest” mean in a particular context? It turns out that having to transmit information of different kinds often results in setting entirely different optimization goals [3], [4]. For example, it is critical to keep the end-to-end delay low in order to ensure proper voice conversation quality, but voice packets are relatively small and do not require much bandwidth. Data transmission, on the other hand, requires a path of the greatest bandwidth in order to minimize the time of transmission, the delay value. However, is often less important and delays at the order of magnitude of seconds or even minutes are acceptable [5]. Besides the bandwidth and end-to-end delay, other link properties such as the number of hops, as well as link reliability, can be also considered. Therefore, traffic classes (eg. voice, video streaming or data) are defined in terms of the link properties, which are known as the Quality of Service (QoS) requirements [3]. Some of these parameters (criteria) must be minimized, whereas others act as constraints which cannot be exceeded [6], [7]. Finding con- strained optimal paths in graphs is an -complete problem and therefore heuristic approaches must be applied to solve it. There are attempts to use abstract algorithms such as genetic algorithms [8], [9] which provide satisfactory results, however their computational complexity limits their use in real network environments. In the paper an alternative algorithm is proposed which is characterized by a low computational complexity. Although it is limited to a subclass of multicriterial routing problems, it may still be applied to solve numerous practical QoS problems. The paper is divided into eight sections. In Section II a multicriterial routing problem has been specified. In Section III the Lagrangian relaxation technique has been presented as a general approach to multicriterial optimization. In Section IV the existing LARAC algorithm has been presented that deals with multicriterial routing but has some limitations overcoming is later presented in this article. In Section V the new algorithm MLARAC as an extension of the LARAC has been presented that enables including multiple constraints to the routing problem. In section VI numerical results have been presented that show example results of MLARAC computa- tions. In Section VII the conclusions have been presented and Section VIII presents future works. II. FORMULATION OF THE PROBLEM We assume that a communication network can be modeled by an undirected graph =(,), where is a finite set of nodes and ⊆{(, ): , } is a set of edges that represent point-to-point links. Furthermore, we assume that every edge is assigned a set of ( + 1) numeric properties (metrics), where is the number of imposed constraints. Met- rics are real-valued functions ( : , =0, 1, ..., ) and reflect the cost of a given edge. For each of the metrics, except the first one, we define a maximum value (constraint) , =1, 2, ..., that cannot be exceeded in the resulting path, this means that the following condition must be satisfied: ∈{1,2,,...,} () , (1) where is a resulting path between given nodes. Given that we define the path’s cost as: ()= 0 (), (2) IEEE Africon 2011 - The Falls Resort and Conference Centre, Livingstone, Zambia, 13 - 15 September 2011 978-1-61284-993-5/11/$26.00 ©2011 IEEE

[IEEE AFRICON 2011 - Victoria Falls, Livingstone, Zambia (2011.09.13-2011.09.15)] IEEE Africon '11 - Lagrangian relaxation in the multicriterial routing

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Page 1: [IEEE AFRICON 2011 - Victoria Falls, Livingstone, Zambia (2011.09.13-2011.09.15)] IEEE Africon '11 - Lagrangian relaxation in the multicriterial routing

Lagrangian Relaxation in the Multicriterial RoutingKrzysztof Stachowiak, Joanna Weissenberg and Piotr Zwierzykowski

Poznan University of Technology, Chair of Communication and Computer Networksul. Polanka 3, 60-965 Poznan, Poland, e-mail: [email protected]

Abstract—Communications networks routing is not a newsubject. However, with a growth of packet networks and theincrease of their range and importance, the task of conveyinginformation over the web has grown beyond its original shapeand has become far more sophisticated. It is no longer computerscientists and the military that make use of the global networkexclusively; it can be stated that almost every possible groupfrom the global population has numerous representatives amongInternet users. This fact is coupled with a decrease in thepopularity of the original communication media such as thetelephone and the e-mail. This is mainly due to them beingassimilated into packet networks along with the process ofnetwork convergence. A number of different types of informationstreams in packet networks has been observed and classified. Itturns out that careful resources management itself may lead toan increase in the information transporting efficiency. In thispaper, an approach is explored - the Lagrangian relaxation - thatenables incorporating multiple criteria into the routing problemsolution. At the same time, the low computation complexity ismaintained that enables further consideration of the techniquealso in practical scenarios.

Index Terms—routing, Lagrangian relaxation, algorithm

I. INTRODUCTION

Communications networks routing is a well known problemthat has been given a lot of attention and some well-developedsolutions have been provided for solving it. Originally, therouting problem has been modeled as the shortest path findinggraph problem which has a good and efficient solution inthe Dijkstra’s algorithm, which is broadly used [1], [2]. Itsmain merit, the simplicity, unfortunately also draws its limits.With communication networks being utilized for more andmore purposes, a differentiation between the so-called classesof traffic emerges. Whereas it is still desired to look for theshortest (optimal) path in the network, it is not always obviouswhat optimization criteria are the most important in a givencase, i.e. what does the “shortest” mean in a particular context?It turns out that having to transmit information of differentkinds often results in setting entirely different optimizationgoals [3], [4]. For example, it is critical to keep the end-to-enddelay low in order to ensure proper voice conversation quality,but voice packets are relatively small and do not require muchbandwidth. Data transmission, on the other hand, requiresa path of the greatest bandwidth in order to minimize thetime of transmission, the delay value. However, is often lessimportant and delays at the order of magnitude of seconds oreven minutes are acceptable [5]. Besides the bandwidth andend-to-end delay, other link properties such as the numberof hops, as well as link reliability, can be also considered.Therefore, traffic classes (eg. voice, video streaming or data)

are defined in terms of the link properties, which are known asthe Quality of Service (QoS) requirements [3]. Some of theseparameters (criteria) must be minimized, whereas others act asconstraints which cannot be exceeded [6], [7]. Finding con-strained optimal paths in graphs is an 𝒩𝒫-complete problemand therefore heuristic approaches must be applied to solve it.There are attempts to use abstract algorithms such as geneticalgorithms [8], [9] which provide satisfactory results, howevertheir computational complexity limits their use in real networkenvironments. In the paper an alternative algorithm is proposedwhich is characterized by a low computational complexity.Although it is limited to a subclass of multicriterial routingproblems, it may still be applied to solve numerous practicalQoS problems.

The paper is divided into eight sections. In Section II amulticriterial routing problem has been specified. In SectionIII the Lagrangian relaxation technique has been presented asa general approach to multicriterial optimization. In SectionIV the existing LARAC algorithm has been presented thatdeals with multicriterial routing but has some limitationsovercoming is later presented in this article. In Section V thenew algorithm MLARAC as an extension of the LARAC hasbeen presented that enables including multiple constraints tothe routing problem. In section VI numerical results have beenpresented that show example results of MLARAC computa-tions. In Section VII the conclusions have been presented andSection VIII presents future works.

II. FORMULATION OF THE PROBLEM

We assume that a communication network can be modeledby an undirected graph 𝐺 = (𝑉,𝐸), where 𝑉 is a finite setof nodes and 𝐸 ⊆ {(𝑢, 𝑣) : 𝑢, 𝑣 ∈ 𝑉 } is a set of edges thatrepresent point-to-point links. Furthermore, we assume thatevery edge is assigned a set of (𝑀 + 1) numeric properties(metrics), where 𝑀 is the number of imposed constraints. Met-rics are real-valued functions (𝑚𝑖 : 𝐸 → ℝ, 𝑖 = 0, 1, ...,𝑀)and reflect the cost of a given edge. For each of the metrics,except the first one, we define a maximum value (constraint)𝑐𝑖 ∈ 𝐶, 𝑖 = 1, 2, ...,𝑀 that cannot be exceeded in the resultingpath, this means that the following condition must be satisfied:

∀𝑖∈{1,2,,...,𝑀}∑

𝑒∈𝑝

𝑚𝑖(𝑒) ≤ 𝑐𝑖, (1)

where 𝑝 is a resulting path between given nodes.Given that we define the path’s cost as:

𝑐(𝑝) =∑

𝑒∈𝑝

𝑚0(𝑒), (2)

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the problem leads to finding a path 𝑝 between nodes 𝑠 and 𝑡(𝑠, 𝑡 ∈ 𝑉 ) such that:

min𝑝∈𝑃 ′(𝑠,𝑡)

𝑐(𝑝), (3)

where 𝑃 ′(𝑠, 𝑡) is a set of all the paths in graph 𝐺 betweennodes 𝑠 and 𝑡 that fulfill condition (1).

III. LAGRANGIAN RELAXATION

In order to solve this problem presented in Section II,a Lagrangian relaxation technique will be used [10]. Theconstraints from Formula (1) are moved into the target func-tion with suitable weights. Therefore, a new target functionreplaces the corresponding function in Formula (3):

min𝑝∈𝑃 ′(𝑠,𝑡)

𝑒∈𝑝

[𝑚0(𝑒) + 𝜆1𝑚1(𝑒) + ...+ 𝜆𝑀𝑚𝑀 (𝑒)] . (4)

The goal of the Lagrangian relaxation is to determine theset of {𝜆1, 𝜆2, ..., 𝜆𝑀} values, that will guarantee keeping theconstraints. However it is also required, that it worsens themain cost of the result (the 𝑚0 metric of the resulting path)the least. In order to achieve that, the relaxation proposesa dual problem which is a maximization of the followingfunction [10]:

𝐿(𝜆1, 𝜆2, ..., 𝜆𝑀 ) = min(∑

𝑒∈𝑝(𝑠,𝑡)

𝑚0(𝑒)+𝑀∑

𝑖=1

𝜆𝑖(𝑚𝑖(𝑒)−𝑐𝑖)). (5)

Maximizing the 𝐿 function gives us the vector𝝀 = {𝜆1, 𝜆2, ..., 𝜆𝑀} which when substituted in Formula (4)satisfies the requirements mentioned above.

Then, the main idea is to redefine the cost of an edge andthe new metric will take on the following form:

𝑐𝜆(𝑒) = 𝑚0(𝑒) +m(𝑒)𝝀𝑇 , (6)

where m(𝑒) is a vector of elements 𝑚1(𝑒),𝑚2(𝑒), ...,𝑚𝑀 (𝑒).The path optimized against 𝑐𝜆 will be a suboptimal solutionto the original problem.

IV. PREVIOUS WORK

The problem of the constrained multicast routing has beenalready approached in several ways. In [11] an interestingnon-linear variant of the lagrangian relaxation is proposed thatpresents one of approaches to the utilization of the relaxationtechnique. In [12] a combination of the Lagrangian relaxationand the branch and bound technique has been proposedas an efficient heuristic to approximate the lagrangian dualsolution. In [13] the routing algorithm, based strictly uponthe Lagrangian relaxation technique, has been proposed. TheLARAC algorithm [13] solves a problem limited to the mainoptimization criterion and only to one constraint. Authorsused some special properties of the optimization problem thatappear when only one constraint is present. This algorithm isa base of the extension proposed in this article.

Fig. 1. Lagrange dual problem visualization for one constraint

A. The LARAC specific problem approach

In such a case, Formula (5) may be depicted using Figure 1.A, B, C and D lines represent the aggregated cost of eachof the feasible paths for a hypothetical problem described byFormula (3) (with constraints left out). The dual Lagrangianproblem for one constraint can be illustrated by increasinglines which represent the paths that exceed the assumedconstraint and by decreasing lines representing the paths thatsatisfy constraint. Let us consider a sample value 𝜆0 (Figure 1).The path B will be chosen as an optimal solution subject to thecost built based on the 𝜆0 weight. For each 𝜆 a different pathmay be interpreted as the optimal one, which has been depictedusing an additional bold line. Notice that this line actuallyrepresents the 𝐿(𝜆) function and that 𝐿 will be a piecewiselinear concave function in its entire domain. If we traversethe 𝐿 function’s domain starting from 𝜆 = 0 towards positiveargument we will be increasing the degree in which the metric𝑚1 affects the aggregated cost (6). It is generally not desirableto increase the 𝜆 weights because it distorts the optimality ofthe solution with regard to the 𝑚0 metric. On the other hand, atlow values of 𝜆 the results break the constraint which rendersthem infeasible. At the maximum of the 𝐿 function a particular𝜆 is achieved with which a feasible solution may be found(indicated by the decreasing aggregated cost function). It isalso the lowest multiplier for which we may obtain a feasiblesolution which means introducing as little involvement of thesecond metric as it is required to achieve.

B. The proposed solution procedure

The LARAC procedure begins with selecting two paths:one optimized against the first metric (further referred toas the cost path) and one optimized against the second one(further referred to as the delay path). They may be perceivedas respectively the path that exceeds the constraint for sureand the one that guarantees satisfying the constraint. It mayactually happen that the cost path satisfies the constraint aswell as that the delay path does not. These special cases meanrespectively an immediate success of the algorithm and an

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immediate failure; supposed they did not occur, the algorithmcontinues. In subsequent steps the straight lines associated withthe two paths are tested for an intersection, and a 𝜆 factor isfound at the intersection point. With this 𝜆 a new path isfound as the shortest path in terms of an aggregated metric(note that this procedure may result in finding different pathsas new lambdas are found in consequent steps). It is thendecided whether to substitute with it the cost or the delaypath which is done in order to narrow the search scope. Afterthe substitution another intersection test is performed (for anew pair of paths), and this continues until the path found forthe new 𝜆 is either the current cost path or the delay path. Itmay be seen in Figure 1, that since the 𝐿 function is concave,this will only happen at the 𝜆 maximizing the 𝐿 function,and this means the end of the algorithm. At each point in theprocedure references to the cost and the delay path are kept,which means that at the end the solution is easily achievableas it is just the current delay path.

V. EXTENDING THE LARAC ALGORITHM - MLARAC

In this paper an extension to LARAC is proposed that allowsincluding more than one constraint to the solved problem. Thismay be advisable due to the fact that in modern networks thereare more properties to describe a single link. Also, there aretraffic classes that may require optimization only against onecriterion while applying constraints to more than one othercriterion.

In a multidimensional approach, the 𝐿 function is still apiecewise linear and concave, but instead of a flat hill likethe one in Figure 1, a multidimensional “hyper-hill” needsto be maximized. Notice the two critical steps in the LARACalgorithm: the choosing of the initial set of paths (the cost andthe delay path), and the second one - the substitution of thenewly found path for one of them. Finding the multidimen-sional equivalents of those is the major problem of extendingthis algorithm for multiple constraints. When introducing addi-tional constraints both of these steps need certain adjustmentsand, unfortunately, loose much of their robustness, which hasbeen shown in the algorithm Multidimensional LARAC.

Algorithm Multidimensional LARAC1. 1: procedure MLARAC(s, t, M, C)

2: 𝐸𝑥𝑃𝑎𝑡ℎ← 𝑆ℎ𝑜𝑟𝑡𝑃𝑎𝑡ℎ(𝑀 [0], 𝑠, 𝑡)3: for 𝑖 := 1𝑡𝑜𝑀 do4: 𝑈𝑛𝑃𝑎𝑡ℎ𝑠[𝑖]← 𝑆ℎ𝑜𝑟𝑡𝑃𝑎𝑡ℎ(𝑀 [𝑖], 𝑠, 𝑡)5: end for6: if 𝑆𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠𝐴𝑙𝑙(𝐸𝑥𝑃𝑎𝑡ℎ) = 𝑇𝑟𝑢𝑒 then7: 𝑅𝑒𝑡𝑢𝑟𝑛𝐸𝑥𝑃𝑎𝑡ℎ8: end if9: for 𝑖 := 1𝑡𝑜𝑀 do

10: if 𝐶𝑜𝑠𝑡𝐹𝑜𝑟𝑀𝑒𝑡𝑟𝑖𝑐(𝑈𝑛𝑃𝑎𝑡ℎ𝑠[𝑖], 𝑖) > 𝑀 [𝑖]then

11: 𝑅𝑒𝑡𝑢𝑟𝑛𝐹𝑎𝑖𝑙𝑢𝑟𝑒12: end if13: end for14: 𝐵𝑒𝑠𝑡𝑈𝑛← 𝐹𝑎𝑖𝑙𝑢𝑟𝑒

15: repeat16: 𝐿𝑎𝑚𝑏𝑑𝑎𝑠← 𝐼𝑛𝑡𝑒𝑟(𝐸𝑥𝑃𝑎𝑡ℎ, 𝑈𝑛𝑃𝑎𝑡ℎ𝑠)17: 𝑃𝑎𝑡ℎ← 𝑆ℎ𝑜𝑟𝑡𝑃𝑎𝑡ℎ(𝐴𝑔𝑔𝑟𝑀(𝐿𝑎𝑚𝑏𝑑𝑎𝑠), 𝑠, 𝑡)18: if 𝑆𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠𝐴𝑙𝑙(𝑃𝑎𝑡ℎ) = 𝑇𝑟𝑢𝑒 then19: 𝐵𝑒𝑠𝑡𝑈𝑛← 𝑃𝑎𝑡ℎ20: end if21: 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒(𝑃𝑎𝑡ℎ,𝐸𝑥𝑃𝑎𝑡ℎ, 𝑈𝑛𝑃𝑎𝑡ℎ𝑠)22: until 𝐴𝑙𝑙𝐴𝑔𝑟𝐶𝑜𝑠𝑡𝑠𝐸𝑞𝑢𝑎𝑙(𝐸𝑥𝑃𝑎𝑡ℎ, 𝑈𝑛𝑃𝑎𝑡ℎ𝑠)23: 𝑅𝑒𝑡𝑢𝑟𝑛𝐵𝑒𝑠𝑡𝑈𝑛24: end procedure

In Line 2 the ExPath path is found, which is the path thatmay potentially exceed all the constraints (in LARAC this isensured as there is only one constraint to be considered). InLines 3 through 5 UnPaths - an array of paths is defined thatsatisfy (“underceed”) at least the constraint respective (basedon their index) to the metric they are associated with. TheExPath summed with UnPaths define the set of the paths thatwill serve as a foundation of finding subsequent approxima-tions of the solution; from now on they will be called theapproximating paths. Note that these are the multidimensionalequivalents of LARAC’s cost and delay paths.

In Line 6 it is checked if the ExPath satisfies all theconstraints and if so, then the ExPath is assumed to be theproblem’s solution and the algorithm breaks successfully. Ifany of the UnPaths exceeds the constraint that it is asso-ciated with (by the fact that it is initially found based onthe respective metric) it means that no path satisfying thisconstraint exists. In such case also no solution exists tothe problem (algorithm’s Lines 9 through 13). So far, theimmediate success and failure conditions have been checkedand the approximating paths set has been defined. If we arestill in the algorithm, it means that an approximation loopmust be entered (Line 15 and further). It is important to notethat in LARAC a feasible solution is already available at thispoint - it is the delay path that satisfies the only constraint inthe problem. However, when considering multiple constraintsthere is no simple and certain way of obtaining a feasiblesolution at this point as each of the UnPaths “ignores” themetrics it is not associated with. Therefore, the BestUn pathappears here and is initially set to a “failure” value as it mayhappen that no feasible solution appears in the following loop.The loop consists of three major steps:

∙ Intersecting the hyperplanes associated with the approx-imating paths in order to find new 𝝀 vector.

∙ Finding a new path candidate, optimal in terms of newaggregated metric shown in Formula (6).

∙ Substituting the new path for one of the approximatingpaths.

The loop breaks, when, for a given 𝝀, vector costs of allapproximating paths are equal. Note that this will only happenwhen the peak of the “hyper-hill” has been found. The threephases of the loop will now be described in detail.

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A. Hyperplanes’ intersection

In Line 16 of the algorithm, the result of the function Inter(meaning intersection) is stored as the vector of lambdas.In order to find the intersection, a proper linear system isbuild. Since there are 𝑀 constraints in this problem, 𝑀lambdas must be found, thus the following system of 𝑀 linearequations is proposed:

𝑀∑𝑖=1

𝜆𝑖(𝑚𝑖𝑢0 −𝑚𝑖𝑒) = 𝑚0𝑒 −𝑚0𝑢0

𝑀∑𝑖=1

𝜆𝑖(𝑚𝑖𝑢1 −𝑚𝑖𝑒) = 𝑚0𝑒 −𝑚0𝑢1

...𝑀∑𝑖=1

𝜆𝑖(𝑚𝑖𝑢𝑀 −𝑚𝑖𝑒) = 𝑚0𝑒 −𝑚0𝑢𝑀

(7)

where

∙ 𝜆𝑖 is the i-th element of the 𝜆 vector,∙ 𝑚𝑖𝑒 means the i-th metric of the “exceeding path”,∙ 𝑚𝑖𝑢𝑗 means the i-th metric of the j-th path from the

“underceeding” paths set.

This system is built based on the assumption that in theintersection point defined by a set of resulting lambdas, theaggregated costs of all of the approximating paths will beequal. The 𝑀 costs associated with the UnPaths form the leftsides of the 𝑀 equations, and the cost of the ExPath is theright side of each of the equations. Formula (7) presents analready simplified form of this system.

B. Finding new solution candidate

In Line 17 of the algorithm the shortest path is found withan aggregated metric as an optimization criterion. In such acase, the cost of inclusion an edge 𝑒 ∈ 𝐸 in the resulting pathtakes the form as in Formula (6). It is worth noting that ifthe path obtained this way satisfies all the constraints, it issubstituted as the current best candidate for the result - theBestUn path.

C. Candidate’s substitution

In Line 21 the newly found path is being substituted forone of the approximating paths. In the LARAC algorithm thissubstitution was easily achievable. If the path exceeded theconstraint (all of the constraints in general), then it had tobe substituted for the cost path, otherwise (i.e. it satisfied allconstraints) it had to be substituted for the delay path. Besidessimplicity this approach has another advantage which sustaina feasible solution all the time, even after the substitution.In the multidimensional optimization, there is no such sharpdiscrimination. The path can of course exceed or satisfy allof the constraints, but it will also occur that it only satisfiessome, while breaking others. A heuristic approach is presentedhereby in order to tackle this problem. If the path breaks allconstraints it is substituted for the “exceeding” path. And ifthe path satisfies some constraints it is substituted for the path

associated with the first constraint that is not broken. Thissubstitution procedure may be adjusted as different heuristicsmay be utilized. However, the general idea stays the same andis about spreading the approximating paths around the “hyper-hill” as even as possible to obtain the set of paths that willpoint towards the hill’s peak most robustly.

VI. NUMERICAL RESULTS

A. Experiment procedure

In order to depict some of the proposed algorithm’s prop-erties, a numerical experiment has been performed. The exactpaths were found using a depth first search algorithm whichwas used to find all possible paths which were in turn searchedfor the lowest cost path that satisfies given constraints. Thisimplied a necessity for only considering relatively small net-works of between 10 and 30 nodes due to the complexity ofrecursive graph search for greater topologies. A set of 1000random graphs was generated, each of which was assigned3 random metrics per each edge. This was done for eachof a number of different node counts (10, 15, 20, 25 and30). Considering small but gradually increasing node countscould serve to observe any emerging trends that would makeit possible to estimate the algorithm’s performance at largernetworks. This may allow to overcome the problem of inabilityto perform a full featured comparison at greater sizes of thetested networks. The distribution of the edge metrics of thegenerated networks was uniform from the (0; 1) range. Thiscan be interpreted as any possible set of metrics normalizedin order to simplify the result’s interpretation and the calibra-tion of the experimental system. Arbitrary constraints whereapplied to the second and the third metric values which werechosen through a calibration process of trial and error in orderto find correct values. They had to be high enough for thefeasible solutions to exist as well as low enough so that theinitial path of the minimal cost (the bandwidth related metric𝑚0) did not satisfy them. This would invalidate the need forperforming approximation steps which were to be observed.

The aim of the experiment was to observe how the usage ofthe heuristic algorithm impacts the quality of the results. Foreach of randomly picked pairs of nodes a path was searchedusing three algorithms. Firstly, a path of minimal cost wasfound with a Dijkstra’s algorithm followed by two searches forconstrained paths. These were performed using respectivelythe exact algorithm and then the MLARAC. The cost of theminimal cost path was then compared with the results obtainedusing the other algorithms. The relative increase of the firstcost in case of the exact solution was compared to the relativeincrease of the respective metric of the result obtained withMLARAC. The rationale behind such a comparison is thatwhen finding a path minimized against the metric 𝑚0 andintroducing constraints to 𝑚1,𝑚2, ... it is expected that the𝑚0 cost of the result may be higher as compared to the resultobtained without the constraint. The exact solution presentsthe smallest possible trade-off (i.e. the additional share of 𝑚0

introduced to “pay” for fulfilling of the constraints), and anyheuristic variant will offer a greater or equal increase. In this

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paper we identify this increase as a measure of the heuristic’squality. Also, an attempt was made to observe some additionalproperties of the algorithm, for larger networks, without thecomparative aspect due to inability of obtaining exact resultsat this level of complexity. However, in that case it was stillpossible to observe the mentioned 𝑚0 cost increase thereforeit has been presented.

B. MLARAC in comparison to exact solution

In Figure 2 the results for the cost increase comparison withthe exact algorithm are presented. In the notation applied inFigure 2 the columns’ denote:

∙ 𝛿𝑒 - the ratio of the cost increase when the exact algorithmwas used for finding the constrained shortest path,

∙ 𝛿𝑚 - the ratio of the cost increase for the MLARACalgorithm.

In the comparison of 𝛿𝑒 and 𝛿𝑚 it can be observed that eventhough MLARAC turns out slightly inferior compared to theexact algorithm, the ratio remains steady. This is promising inobtaining results close to optimal results also for bigger net-works. It can also be seen that not often many approximationswere needed, which highlights the good convergence of thealgorithm.

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

10 15 20 25 30

Cos

t inc

reas

e

Number of network nodes

δe δm

Fig. 2. Comparative cost increase

In Figure 3 the average number of approximation steps thatwere necessary to obtain a result is shown. It can be seen thatnot often many approximations were needed, which highlightsthe good convergence of the algorithm.

C. MLARAC performance for big structures

A similar procedure has been performed for bigger graphs.For each of the set of node numbers a set of 1000 graphswas processed. In each graph a path between a random pairof nodes was searched first with Dijkstra’s algorithm forobtaining of the minimal path’s cost and then with MLARACto record the cost increase in the same fashion as it was donefor the smaller graphs.

In Figure 4 the results of the cost increase are shown,whereas Figure 5 shows average numbers of the approximationsteps.

1

1.02

1.04

1.06

1.08

1.1

1.12

10 15 20 25 30

Avg

. num

ber

of a

ppro

xim

atio

ns

Number of network nodes

Fig. 3. Average number of approximations for small networks

1.18

1.185

1.19

1.195

1.2

1.205

1.21

100 200 300 400

Cos

t inc

reas

e

Number of network nodes

Fig. 4. Cost increase for big networks

1.3

1.31

1.32

1.33

1.34

1.35

1.36

1.37

1.38

1.39

100 200 300 400

Avg

. num

ber

of a

ppro

xim

atio

ns

Number of network nodes

Fig. 5. Aferage number of approximations for big networks

It can be seen that the cost increase recorded for thebigger networks is similar to the results from the previouscomparison. However, the average number of approximationsincreased slightly, which may be explained by the fact that inbigger topologies there are more possible paths to be exploredbefore a decision to accept a certain result. Also an interestingrelation between the number of approximations and the costincrease may be observed. This trend that is common todifferent properties may be viewed as a rough estimate of thealgorithm’s complexity in the function of the number of the

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networks’ nodes.

VII. CONCLUSION

Multicriterial routing has still not been throughly exploredyet, and Lagrangian relaxation is one of the fields that havenot yet been fully covered in this particular use. The previousexperiments as well as those presented in this paper showthat it may serve the QoS routing issue. Some degrees offreedom in the Lagrangian relaxation have been shown, suchas the choice of the algorithm of maximizing the piecewiselinear function from the Lagrangian dual problem. Among theaspects of the particular algorithm presented a very vital partis the sub-procedure referred to as the path substitution. In thefuture, some more specific approaches may be proposed inthis field. One of the more promising is approximating pathsby the sub-gradients associated with their hyperplanes and tryto assume a relation between the sub-gradient’s value and thecloseness to the hill’s peak. This approach may be extended byconcluding from the entire approximation loop’s history andtake a form of a swarm intelligence based algorithm. Alsoa comparison may be made with a set of simple randomlychosen strategies leading to finding the best one based on thetrial and error approach. Care must be taken when assumingfrom the Lagrangian relaxation utilization on other fields.Thing as they are, it often happens that a specific problemoptimization to which a general technique, such as Lagrangianrelaxation, is being applied, creates a unique context in whicha given algorithm reveals some new properties.

VIII. FUTURE WORKS

Further studies will be performed in search for such specialcharacteristic in the Lagrangian relaxation for multicriterialnetwork routing as the initial results have turned out promis-ing. Therefore the next stage in the authors’ research workaiming to implementation of MLARAC in large networksand then application the proposed algorithm for the multicastrouting problem in order to obtain the shortest path tree [14].

REFERENCES

[1] J. Moy, “RFC 1583: OSPF version 2,” March 1994.[2] D. Oran, “OSI IS-IS Intra-domain Routing Protocol,” Feb. 1990.[3] M. Peuhkuri, “IP Quality of Service,” Helsinki University of Technology,

Laboratory of Telecommunications Technology, Tech. Rep., 1999.[4] Y. Jiang and J. Wu, “nPFS: Efficient management of multiple QoS

objects,” in Proceedings of the 2001 International Conference on Com-puter Networks and Mobile Computing (ICCNMC’01), ser. ICCNMC’01. Washington, DC, USA: IEEE Computer Society, 2001, p. 49.

[5] A. Ranjbar, CCNP ONT Official Exam Certification Guide, ser. ExamCertification Guide. Cisco Press, May 2007.

[6] M. Piechowiak and P. Zwierzykowski, “A New Delay-ConstrainedMulticast Routing Algorithm for Packet Networks,” in Proceeding ofIEEE Africon, Nairobi, Kenya, 2009.

[7] M. Piechowiak, “Evaluation of Heuristic Algorithms for Multicast Con-nections in Packet Networks,” Ph.D. dissertation, Poznan University ofTechnology, Chair of Communication and Computer Networks, Poznan,Poland, 2010, in polish.

[8] L. Barolli, A. Koyama, H. Sawada, T. Suganuma, and N. Shiratori, “Anew QoS routing approach for multimedia applications based on geneticalgorithms,” in Proceedings of the the First International Conference onCyberworlds. Los Alamitos, CA, USA: IEEE Computer Society, 2002,p. 0289.

[9] A. Chojnacki, M. Piechowiak, and P. Zwierzykowski, “Genetic RoutingAlgorithm for Multicast Connections in Packet Networks,” InternationalJournal of Image Processing & Communications, vol. 13, no. 1-2, pp.13–20, 2008.

[10] R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network Flows : Theory,Algorithms, and Applications. Englewood Cliffs, NJ: Prentice Hall,1993.

[11] G. Feng, “The revisit of nonlinear lagrange relaxation based qos rout-ing,” in Communications, 2005. ICC 2005. 2005 IEEE InternationalConference on, vol. 1, may 2005, pp. 196 – 200 Vol. 1.

[12] H. Jiang, P. Yan, J. Zhou, L. Chen, and M. Wu, “Multi-constrained leastcost qos routing algorithm,” in Telecommunications and Networking- ICT 2004, ser. Lecture Notes in Computer Science, J. de Souza,P. Dini, and P. Lorenz, Eds. Springer Berlin / Heidelberg, 2004,vol. 3124, pp. 704–710, 10.1007/978-3-540-27824-5_94. [Online].Available: http://dx.doi.org/10.1007/978-3-540-27824-5_94

[13] A. Juttner, B. Szviatovszki, I. Mecs, and Z. Rajko, “Lagrange RelaxationBased Method for the QoS Routing Problem,” in Proceedings of 12thAnnual Joint Conference of the IEEE Computer and CommunicationsSocieties INFOCOM’2001, vol. 2, Anchorage, USA, 2001, pp. 859–868.

[14] M. Piechowiak and P. Zwierzykowski, “Efficiency Analysis of MulticastRouting Algorithms in Large Networks,” in Proceedings of The ThirdInternational Conference on Networking and Services ICNS 2007, IEEE.Athens, Greece: IEEE, June 2007, pp. 101–106, the best paper award.

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