QoS Routing Under Multiple Additive Constraints: A ...optimization.asu.edu/papers/XUE-JNL-2016-TETC-V4-P242.pdf · IEEE TRANSACTIONS ON EMERGING TOPICS IN COMPUTING Xiao et al.: QoS

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EMERGING TOPICSIN COMPUTING

Received 30 November 2014; revised 9 February 2015; accepted 27 April 2015.Date of Publication 5 May 2015; date of current version 8 June 2016.

Digital Object Identifier 10.1109/TETC.2015.2428654

QoS Routing Under Multiple AdditiveConstraints: A Generalization

of the LARAC AlgorithmYING XIAO1, KRISHNAIYAN THULASIRAMAN2, (Fellow, IEEE),

GUOLIANG XUE3, (Fellow, IEEE), AND MAMTA YADAV2, (Student Member, IEEE)1VT iDirect Inc., Herndon, VA 20171 USA

2School of Computer Science, University of Oklahoma, Norman, OK 73019 USA3School of Computing, Informatics and Decision Systems Engineering Arizona State University, Tempe, AZ 85287 USA

CORRESPONDING AUTHOR: K. THULASIRAMAN ([email protected])

The work of K. Thulasiraman was supported by the National Science Foundation (NSF) Information Technology Researchunder Grant ANI-0312435. The work of G. Xue was supported in part by the U.S. Army Research Office

under Grant W911NF-09-1-0467 and in part by NSF under Grant 1217611 and Grant 1421685.

ABSTRACT Consider a directed graph G(V ,E), where V is the set of nodes and E is the set oflinks in G. Each link (u, v) is associated with a set of k + 1 additive nonnegative integer weightsCuv = (cuv,w1

uv,w2uv, . . . ,w

kuv). Here, cuv is called the cost of link (u, v) and wiuv is called the ith delay

of (u, v). Given any two distinguish nodes s and t , the QoS routing (QSR) problem QSR(k) is to determinea minimum cost s − t path such that the ith delay on the path is atmost a specified bound. This problemis NP-complete. The LARAC algorithm based on a relaxation of the problem is a very efficient approx-imation algorithm for QSR(1). In this paper, we present a generalization of the LARAC algorithm calledGEN-LARAC. A detailed convergence analysis of GEN-LARACwith simulation results is given. Simulationresults provide an evidence of the excellent performance of GEN-LARAC.We also give a strongly polynomialtime approximation algorithm for the QSR(1) problem.

INDEX TERMS QoS routing, Lagrangian relaxation, combinatorial optimization, algorithms.

I. INTRODUCTIONShortest path, minimum cost flow, and maximum flow com-putation are fundamental problems in operations research.Though interesting in their own right, algorithms for theseproblems also serve as building blocks in the design ofalgorithms for complex problems encountered in large scaleindustrial applications. Hence, over the years there has beenan extensive literature on various aspects of these problems.These problems are solvable in polynomial time. But addingone or more additional additive constraints makes themintractable.

In this paper, we focus on the constrained shortestpath (CSP) problem also known as the QoS routing (QSR)problem. This problem requires determination of a mini-mum cost path from a source node to a destination nodeof a network subject to bounds on certain additive metricson the path. For instance, the additive metric commonlyconsidered is the total delay of the path. The QoS problem

has attracted considerable attention from different researchcommunities: operations research, computer science, andtelecommunication networks. The interest from the telecom-munications community arises from the great deal of empha-sis on the need to design communication protocols thatdeliver certain performance guarantees. This need, in turn,is the result of an explosive growth in high bandwidth realtime applications that require stringent quality of serviceguarantees.

A. REVIEW OF LITERATUREThe literature on the QoS routing problem is vast. Hencein this subsection we survey only a subset of publishedworks in this area. We first review the literature on theQoS routing problem under one additive metric followedby the literature on several additive metrics and othergeneralizations.

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Xiao et al.: QoS Routing Under Multiple Additive Constraints



1) QoS ROUTING PROBLEM UNDER ONEADDITIVE METRICIt has been shown in [5], [29] that the QoS routing problemis NP-hard even for acyclic networks. So, in the literature,heuristic approaches and approximate algorithms have beenproposed. Heuristics, in general, do not provide performanceguarantees on the quality of the solution produced, thoughthey are usually fast in practice. On the other hand,ε-approximation algorithms deliver solutions with costwithin (1+ ε) time the optimal cost, but are usually very slowin practice because they guarantee the quality of the solutions.Hassin [7] gave the first ε− approximation algorithm for theQoS routing problem under one additive metric.

As regards heuristics, a number of them have appeared inthe literature providing different levels of performance withregard to the quality of the solution as well as the computationtime required. For instance, the LHWHM algorithm [17] isa very simple heuristic which is very fast requiring onlytwo invocations of Dijkstra’s shortest path algorithm for afeasible problem. Reference [25] also discusses furtherenhancements of the LHWHM algorithm as well as a heuris-tic based on the Bellman-Ford-Moore (BFM) algorithm forthe shortest path problem. It should be emphasized that inall these cases, only simulations are used to evaluate theperformance of the algorithms. A comprehensive overviewof a number of QoS routing algorithms may be found in [16].

There are heuristics that are based on sound theoreticalfoundations. These algorithms are based on solutions to theinteger relaxation or the dual of the integer relaxation of theQoS routing problem. To the best of our knowledge, the firstsuch algorithm was reported in [6] by Handler and Zhang.This is based on the geometric approach (also called thehull approach [19], [41]). More recently, in an independentwork, Jüttner et al. [9] developed the LARAC algorithmwhich solves the Lagrangian relaxation of the QoS routingproblem. In another independent work, Blokh and Gutin [3]defined a general class of combinatorial optimization prob-lems (that are called the MCRT problems, namely, MinimumCost Restricted Time Combinatorial Optimization problems)of which the QoS routing problem is a special case, andproposed an approximation algorithm to this problem.Xiao et al. [32] drew attention to the fact that the algorithmsin [3] and [9] are equivalent. Mehlhorn and Ziegelmann [19]and Ziegelmann [41] have also observed this equivalenceand have developed several insightful results. In viewof this equivalence, we refer to these algorithms as theLARAC algorithm. The work in [32] also establishes cer-tain results using the algebraic approach. These results alsohold true in the case of the general optimization problemconsidered in [6]. In another independent work, Xue [39]also arrived at the LARAC algorithm using the primal-dual method of linear programming. Usually, approximationalgorithms are developed using the dual of the relaxed versionof the QoS routing problem. Xiao et al. [33] described anefficient approximation algorithm for the QoS routing prob-lem using the primal simplex method of linear programming.

In [10], Jüttner proved the strong polynomiality of theLARAC algorithm, both for the general case and for the QoSrouting problem. He has used certain results from the generalarea of fractional combinatorial optimization. Jüttner [11]gave a general method to solve budgeted optimizationproblems in strongly polynomial time. Radzik [24] gives anexcellent exposition of approaches to fractional combinato-rial optimization problems.

B. MULTI-CONSTRAINED AND DISJOINT PATH ROUTINGAn application of the parametric search method to the generalclass of combinatorial optimization problems involvingtwo additive parameters may be found in [12].Multi-constrained routing problem (routing under severalmetrics) has been considered in [8], [13], [15], [37]–[40].Several interesting algorithms related to the QoS routingproblem and motivated by applications have appeared inthe literature. The constrained disjoint shortest path problemhas been studied in [28] and [32]. [2], [4], [20] and [26]discussed issues involving reliability and hop constraints,[21] and [22] introduced the concept of exact QoS routingand tunability, and [14], [30], [34], [36] discussed constraineddisjoint paths problem, routing under uncertainty and relatedapproximation schemes.

Summarizing the LARAC algorithm and other variants ofthe mathematical programming based approach have beenreported for the QoS routing problem under one additivemetric. But no generalization of the LARAC algorithm appli-cable to multiple additive metric has been studied. In viewof the highly efficient performance of the LARAC algorithmsuch a generalization is of great interest. In this paper wepresent such a generalization. This is an edited version of ourearlier conference paper [35].

The paper is organized as follows. In Section II, we reviewthe QoS routing problem and the LARAC algorithm of [9]for this problem. In Section III we develop GEN-LARAC,a generalization of the LARAC algorithm for the QoS routingproblem with multiple additive constraints. In this section wealso develop a strongly polynomial time algorithm for theinteger relaxation of the special case of one additive metric.A detailed analysis of the convergence properties is also givenwith simulation results providing evidence of the excellentperformance of GEN-LARAC.

II. QoS ROUTING (QSR) PROBLEM ANDTHE LARAC ALGORITHMFirst we give a formal definition of the QoS routing problem.We use the terms links and nodes for edges and vertices,respectively, following the convention in the networkingliterature.

A. QoS Routing Problem (QSR)Consider a network G(V ,E). Each link (u, v) ∈ E isassociated with two weights cuv > 0 (say, cost) andduv > 0 (say, delay). Also are given two distinguished nodess and t and a real number1 > 0. Let Pst denote the set of all

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EMERGING TOPICSIN COMPUTING Xiao et al.: QoS Routing Under Multiple Additive Constraints

s− t paths and for any s− t path p, define

c(p) =∑

(u,v)∈p

cuv and d(p) =∑

(u,v)∈p

duv.

Let Pst (1) be the set of all the s − t paths p such thatd(p) ≤ 1. A path in the set Pst (1) is called a feasible path.The QSR problem is to find a path p∗ = arg min{c(p)|p ∈Pst (1)}. In other words, the QSR problem is to find a min-imum cost feasible s − t path. It can be formulated as thefollowing integer linear program.

QSR :Minimize

∑(u,v)∈E

cuvxuv

subject to ∀u ∈ V ,∑{v|(u,v)∈E}

xuv −∑

{v|(v,u)∈E}

xvu =

1, for u = s−1, for u = t0, otherwise∑

(u,v)∈E

−duv · xuv − w = −1, w ≥ 0

xuv = 0 or 1, ∀(u, v) ∈ E

The QSR problem is known to be NP-hard [5], [29]. Themain difficulty lies with the integrality condition that requiresthat the variables xuv be 0 or 1. Removing or relaxing thisrequirement from the above integer linear program and lettingxuv ≥ 0 leads to RELAX-QSR, the relaxedQSR problem. It isoften convenient to solve the dual of the relaxed form of theQSR problem which we present below.The dual involves s − t paths and a variable λ ≥ 0.

For each link (u, v), let the aggregated cost cλ be defined ascuv+λduv. For a given λ, let cλ(p) denote the aggregated costof the path p. Finally define L(λ) as:

L(λ) = min{cλ(p)|p ∈ Pst } − λ1. (1)

Note that in the above, min{cλ(p)|p ∈ Pst } is the same asthe minimum aggregated cost of an s − t path with respectto a given value of λ. This can be easily obtained by apply-ing Dijkstra’s algorithm using aggregated link costs. Let thes−t path which has minimum aggregated cost with respect toa given λ be denoted as pλ. Then L(λ) = cλ(pλ)−λ1 and thedual of the RELAX-QSR can be presented in the followingform.

DUAL− RELAX−QSR : Find L∗ = max{L(λ)|λ ≥ 0}.

We note that the problem of maximizing L(λ) as aboveis also called the Lagrangian dual problem. The value of λthat achieves the maximum L(λ) in DUAL-RELAX-QSRwill be denoted by λ∗. Note that L∗, the optimum value ofDUAL-RELAX-QSR is a lower bound on the optimum costof the path solving the corresponding QSR problem. The keyissue in solving DUAL-RELAX-QSR is how to search for theoptimal λ and determining the termination condition for thesearch. The LARAC algorithm of [9] presented in Figure 1 isone such efficient search procedure.

FIGURE 1. LARAC algorithm.

B. Description of the AlgorithmIn the LARAC algorithm Dijkstra(s, t, c), Dijkstra(s, t, d)and Dijkstra(s, t, cλ) denote, respectively, Dijkstra’s shortestpath algorithm using link costs, link delays, and aggregatedlink costs with respect to the multiplier λ.1. In the first step, the algorithm calculates the shortest

path on link costs. If the path found meets the delayconstraint, this is surely the optimal path. Otherwise, thealgorithm stores the path as the latest infeasible path,simply called the pc path. Then it determines the shortestpath on link delays denoted as pd . If pd is infeasible,there is no solution to this instance.

2. Set λ = (c(pc) − c(pd ))/(d(pd ) − d(pc)). With thisvalue of λ, we can find a new cλ-minimal path r .If cλ(r) = cλ(pc) = cλ(pd )), we have obtained theoptimal λ as proved in [9] and [32]. Otherwise, set r asthe new pc or pd according to whether r is infeasible orfeasible.

An algebraic study of the LARAC algorithm and an effi-cient implementation called the LARAC-BIN algorithm hasbeen presented in [32].

III. GEN-LARAC: A GENERALIZED APPROACH TO THECONSTRAINED SHORTEST PATH PROBLEM UNDERMULTIPLE ADDITIVE CONSTRAINTSIn this section we study the QSR(k) problem that requiresdetermination of s − t paths that satisfy k > 1 additiveconstraints. We develop a new approach using Lagrangianrelaxation. We use the LARAC algorithm as a building blockin the design of this approach.

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A. FORMULATION OF THE QSR(K) PROBLEMAND ITS RELAXATIONConsider a directed graphG(V ,E) whereV is the set of nodesand E is the set of links in G. Each link (u, v) is associatedwith a set of k + 1 additive non-negative integer weightsCuv = (cuv,w1

uv,w2uv, . . . ,w

kuv). Here cuv is called the cost

of link (u, v) and wiuv is called the ith delay of (u, v). Giventwo nodes s and t , an s− t path in G is a directed simple pathfrom s to t . Let Pst denote the set of all s− t paths in G. Foran s− t path p define

c(p) ≡∑

(u,v)∈p

cuv and di(p) ≡∑

(u,v)∈p

wiuv, i = 1, . . . , k.

The value c(p) is called the cost of path p, and di(p) is calledthe ith delay of path p. Given k positive integers r1, r2, . . . , rk ,an s − t path is called feasible (resp. strictly feasible) ifdi(p) ≤ ri(resp. di(p) < ri), for all i = 1, 2, . . . , k(ri is called the bound on the ith delay of a path).

The QSR(k) problem is to find a minimum cost s − tpath, such that for all i, the ith delay of the path is at most aspecified upper bound ri. An instance of the QSR(k) problemis strictly feasible if all the feasible paths are strictly feasible.Without loss of generality, we assume that the problem underconsideration is always feasible. In order to guarantee strictfeasibility, we do the following transformation.

For i = 1, 2, . . . , k , transform the ith delay of each link(u, v) such that the new weight vector C ′uv is given by

C ′uv = (cuv, 2w1uv, 2w

2uv, . . . , 2w

kuv).

Also transform the bounds r ′i s so that the new vector ofbounds R′ is given by

R′ = (2r1 + 1, 2r2 + 1, . . . , 2rk + 1).

In the rest of the section, we only consider the trans-formed problem. Thus all link delays are even integers,and delay bounds are odd integers. We will use symbolswith capital or bold letters to represent vectors. Also, fora matrix A, AT denotes its transpose. For simplicity ofpresentation, we will use Cuv and R instead of C ′uv and R′

to denote the transformed weight vector and the vector ofbounds.

Two immediate consequences of this transformation arestated below.Lemma 1: ∀ p ∈ Pst , ∀ i ∈ {1, 2, . . . , k}, di(p) 6= ri in the

transformed problem.Lemma 2: An s− t path in the original problem is feasible

(resp. optimal) iff it is strictly feasible (resp. optimal) in thetransformed problem.

Starting with an ILP (Integer Linear Programming)formulation of the QSR(k) problem and relaxing theintegrality constraints we get the RELAX-QSR(k) prob-lem below. In this formulation, for each s − t path p,

we introduce a variable xp.

RELAX−QSR(k):

Minimize∑p

c(p)xp (2)

subject to∑p

xp = 1 (3)∑p

di(p)xp ≤ ri, i = 1, . . . , k (4)

xp ≥ 0, ∀ p ∈ Pst (5)

The Lagrangian dual of RELAX-QSR(k) is given below.

DUAL− RELAX−QSR(k):

Maximize w− λ1r1 · · · − λkrk (6)

subject to w− d1(p)λ1 · · · − dk (p)λk ≤ c(p),

∀p ∈ Pst (7)

λi ≥ 0, i = 1, . . . , k (8)

In the above dual problem λ1, λ2, . . . , λk and w are thedual variables, with w corresponding to (3) and each λicorresponding to the ith constraint in (4).

It follows from (7) that w ≤ c(p) + d1(p)λ1 + · · · +dk (p)λk ,∀p ∈ Pst . Since we want to maximize (6), the valueof w should be as large as possible, i.e.

w = minp∈Pst {c(p)+ d1(p)λ1 + · · · + dk (p)λk}.

With the vector3 defined as3 = (λ1, λ2, . . . , λk ), define

L(3) = minp∈Pst {c(p)+ λ1(d1(p)− r1)

+ · · · + λk (dk (p)− rk )}. (9)

Notice that L(3) is called the Lagrangian function in theliterature and is a concave continuous function of 3.Then DUAL-RELAX-QSR(k) can be written as follows.

DUAL− RELAX−QSR(k):

Maximize L(3)

subject to 3 ≥ 0 (10)

The 3∗ that maximizes (10) is called the maximizingmultiplier and is defined as

3∗ = arg max3≥0L(3) (11)

Claim 1: If an instance of the QSR(k) problem isfeasible and a path popt is an optimal path, then ∀3 ≥ 0,L(3) ≤ c(popt ).We shall use L(3) as a lower bound of c(popt ) to evalu-

ate the quality of the approximate solution obtained by ouralgorithm. Given p ∈ Pst and 3, define

C(p) ≡ (c(p), d1(p), d2(p), . . . , dk (p)),

D(p) ≡ (d1(p), d2(p), . . . , dk (p)),

R ≡ (r1, r2, . . . , rk ),

c3(p) ≡ c(p)+ d1(p)λ1 + · · · + dk (p)λk , and

d3(p) ≡ d1(p)λ1 + · · · + dk (p)λk .




Here c3(p) and d3(p) are called the aggregated cost and theaggregated delay of path p, respectively. We shall use P3 todenote the set of s−t paths attaining the minimum aggregatedcost with respect to3. A path p3 ∈ P3 is called a3-minimalpath.

B. A STRONGLY POLYNOMIAL TIME APPROXIMATIONALGORITHM FOR QSR(1) PROBLEMThe key issue now is to search for the maximizing multiplierand determine termination conditions. If there is only onedelay constraint, i.e., k = 1, we have the following claimfrom [9], [32] repeated below for ease of reference.Theorem 1: Avalue λ > 0 maximizes the function L(λ)

if and only if there are paths pc and pd which are bothcλ-minimal and for which d(pc) ≥ r and d(pd ) ≤ r. (pc andpd can be the same. In this case d(pd ) = d(pc) = r).Theorem 2: DUAL-RELAX-QSR(1) is solvable in

O((m+ n log n)2) time.Proof: We prove this theorem by presenting an

algorithm with O((m + n log n)2) time complexity.The algorithm called PSQSR (parametric search basedQoS routing) algorithm is based on a methodology firstproposed by Megiddo [18] to solve fractional combinatorialoptimization problems.

FIGURE 2. PSQSR algorithm for QSR(1) problem.

Assume node 1 is the source and node n is the target.In Figure 2, we present algorithm PSQSR for computinga QoS routing using lexicographic order on a pair of linkweights (luv, cuv)∀(u, v) ∈ E and based on parametricsearch, where luv = cuv + λ∗duv and λ∗ is unknown. Thealgorithm is the same as the BFM (Bellman-Ford-Moore)algorithm except for Step 4 which needs special care. We useBFM algorithm here because it is easy to explain. Actually weuse Dijkstra’s algorithm for better time complexity results.

In Figure 2, we need extra steps (Oracle test) to evaluate theBoolean expression in the if statement in Step 4 since λ∗ ≥ 0is unknown. If xv = ∞, yv = ∞, then the inequality holds.Assume xv and yv are finite (non-negative) values. Then itsuffices to evaluate the following Boolean expression.

p+ qλ∗ ≤ 0?, where p = xu + cuv − xv and

q = (yu + duv − yv).

FIGURE 3. Oracle test algorithm.

If p · q ≥ 0, then it is trivial to evaluate the Booleanexpression. Without loss of generality, assume p · q < 0,i.e., −p/q > 0. The Oracle test algorithm is presentedin Figure 3.

The time complexity of the Oracle test is O(m + n log n).On the other hand, we can revise the algorithm in Figure 2using Dijkstra’s algorithm and the resulting algorithm willhave time complexity O((m+ n log n)2).Next, we show how to compute the value of λ∗ and L(λ∗).

The algorithm in Figure 2 computes a λ∗-minimal path pwith minimal cost. Similarly, we can compute a λ∗-minimalpath q with minimal delay. Then the value of λ∗ is given bythe following equation: c(p) + λ∗d(p) = c(q) + λ∗d(q) andL(λ∗) = c(p) + λ∗(d(p) − D), where D is the path delayconstrain (here k = 1). Notice that d(q) 6= d(p) is guaranteedby Theorem 1 and the transformation in Section III.A. �

Because PSQSR and LARAC algorithms are based on thesamemethodology and obtain the same solution, we shall alsocall our algorithm LARAC. In the rest of the paper, we shalldiscuss how to extend it for k > 1. In particular we develop anapproach that combines the LARAC algorithm as a buildingblock with certain techniques in mathematical programming.We shall call this new approach as GEN-LARAC.

C. GEN-LARAC FOR THE QSR(K) PROBLEM1) OPTIMALITY CONDITIONSTheorem 3: Given an instance of a feasible QSR(k) prob-

lem, a vector 3 ≥ 0 maximizes L(3) if and only iff thefollowing problem in the variables uj is feasible.∑

pj∈P3

uj · di(pj) = ri, ∀i, λi > 0 (12)





∑pj∈P3

uj · di(pj) ≤ ri, ∀i, λi = 0 (13)

∑pj∈P3

uj = 1, (14)

uj ≥ 0, ∀pj ∈ P3 (15)Proof:

Sufficiency: Let x = (u1, . . . , ur , 0, 0, . . . ) be a vector ofsize |Pst |, where r = |P3|. Obviously, x is a feasible solutionto RELAX-QSR(k). It suffices to show that x and 3 satisfythe complementary slackness conditions.

According to (7), ∀p ∈ Pst ,w ≤ c(p) + d1(p)λ1 +· · · + dk (p)λk . Since we need to maximize (6), the optimalw = c(p3) + d1(p3)λ1 + · · · + dk (p3)λk ,∀p3 ∈ P3. Forall other paths p,w − c(p) + d1(p)λ1 + · · · + dk (p)λk < 0.Hence x satisfies the complementary slackness conditions.By (12) and (13), 3 also satisfies complementary slacknessconditions.Necessary: Let x∗ and (w,3) be the optimal solution to

RELAX-QSR(k) and DUAL-RELAX-QSR(k), respectively.It suffices to show that we can obtain a feasible solution to(12)-(15) from x∗.

We know that all the constraints in (7) correspond-ing to paths in Pst − P3 are strict inequalities, andw = c(p3) + d1(p3)λ1 + · · · + dk (p3)λk ,∀p3 ∈ P3.Therefore, from complementary slackness conditions we getxp = 0,∀p ∈ Pst − P3.

Now let us set uj corresponding to path p in P3 equal to xp,and set all other uj’s corresponding to paths not in P3 equal tozero. The ui’s so elected will satisfy (12) and (13) since theseare complementary conditions satisfied by (w,3). Since xi’ssatisfy (3), uj’s satisfy (14). Thuswe have identified a solutionsatisfying (12)-(15). �

2) GEN-LARAC: A COORDINATE ASCENT METHODOur approach for the QSR(k) problem is based on the coordi-nate ascentmethod in Figure 4 and proceeds as follows. Givena multiplier 3, in each iteration we try to improve the valueof L(3) by updating one component of the multiplier vector.If the objective function is not differentiable, the coordinateascent method may get stuck at a corner 3s not being ableto make progress by only changing one component. We call3s a pseudo optimal point which requires updates of at leasttwo components to achieve improvement in the solution.We shall discuss later how to jump to a better solution from apseudo optimal point. Our simulations show that the objectivevalue attained at pseudo optimal points is usually very closeto the maximum value of L(3).

3) VERIFICATION OF OPTIMALITY OF 3In Step 3 we need to verify if a given 3 is optimal.We show that this can be accomplished by solvingthe following LP (Linear Programming) problem, whereP3 = {p1, p2, . . . , pr } is the set of 3-minimal paths.

Maximize 0 (16)

subject to∑pj∈P3

uj · di(pj) = ri, ∀i, λi > 0 (17)

FIGURE 4. GEN-LARAC: A coordinate ascent algorithm for theQSR(k) problem.

∑pj∈P3

uj · di(pj) ≤ ri, ∀i, λi = 0 (18)

∑pj∈P3

uj = 1, (19)

uj ≥ 0, ∀pj ∈ P3 (20)

By Theorem 3, if the above linear program is feasible thenthe multiplier 3 is a maximizing multiplier.Let (y1, . . . , yk , δ) be the dual variables corresponding to

the above problem. Let Y = (y1, y2, . . . , yk ). The dualof (17)-(20) is as follows

Minimize RY T + δ (21)

subject to D(pi)Y T + δ ≥ 0, i = 1, 2, . . . , r (22)

yi ≥ 0, ∀i, λi > 0 (23)

Evidently the LP problem (21)-(23) is feasible. From therelationship between primal and dual problems, it follows thatif the linear program (16)-(20) is infeasible, then the objectiveof (21) is unbounded (−∞). Thus, if the optimum objectiveof (21)-(23) is 0, then the linear program (16)-(20) is feasibleand by Theorem 3 the corresponding multiplier3 is optimal.In summary, we have the following lemma.Lemma 3: If (16)-(20) is infeasible, then ∃Y =

(y1, y2, . . . , yk ) and δ satisfying (22)-(23) and RY T + δ < 0.The Y in Lemma 3 can be identified by applying any

LP solver on (21)-(23) and terminating it once the currentobjective value becomes negative.

Let 3 be a non-optimal Lagrangian multiplier and3(s,Y ) = 3+ Y/s for s > 0.




Theorem 4: If a multiplier 3 ≥ 0 is not optimal, then

∃M > 0, ∀s > M , L(3(s,Y )) > L(3).Proof: If M is big enough, P3 ∩ P3(s,Y ) 6= φ.

Let pj ∈ P3 ∩ P3(s,V ).

L(3(s,Y )) = c(pj)+ (D(pj)− R)(3+ Y/s)T

= c(pj)+ (D(pj)− R)3T+(D(pj)−R)(Y/s)T

= L(3)+ (D(pj)Y T − RY T )/s.

SinceD(pj)Y T+δ ≥ 0 andRY T+δ < 0,D(pj)Y T−RY T > 0.Hence L(3(s,Y )) > L(3). �We can find the proper value of M by binary search after

computing Y . The last issue is to compute P3. It can beexpected that the size of P3 is usually very small. In ourexperiments, |P3| never exceeded 4 even for large and densenetworks. The k-shortest path algorithm can be adapted easilyto computing P3.

D. ANALYSIS OF THE ALGORITHMIn this section, we shall discuss the convergence properties ofGEN-LARAC.Lemma 4: If there is a strictly feasible path, then for any

given τ , the set Sτ = {3|L(3) ≥ τ } ⊂ Rk is bounded.Proof: Let p∗ be a strictly feasible path. For any

3 = (λ1, . . . , λk ) ∈ Sτ , we have

c(p∗)+ λ1(d1(p∗)−r1)+ · · ·+ λk (dk (p∗)− rk ) ≥ L(3) ≥ τ.

Since di(p∗)− ri < 0 and λi ≥ 0 for i = 1, 2..., k,3must bebounded. �By Theorem 1, we have the following lemma.Lemma 5: A multiplier 3 ≥ 0 is pseudo optimal if∀i ∃pic, p

id ∈ P3, di(p

ic) ≥ ri and di(p

id ) ≤ ri

For an n-vector V = (v1, v2, . . . , vn), let |V |1 = |v1| +|v2| + · · · + |vn| denote the L1-norm.Lemma 6: Let3 and H be two multipliers obtained in the

same while-loop in Step 2 in Figure 4. Then |L(H )−L(3)| ≥|H −3|1.

Proof: Let 3 = 31,32, . . . , 3j= H be the consecu-

tive sequence of multipliers obtained in Step 2. We first showthat |L(3i+1)− L(3i)| ≥ |3i+1

−3i|1.

Consider two cases.

Case 1: λi+1b > λib.By Theorem 1 and Lemma 1,

∃p3i+1 and db(p3i+1 ) > rb.

By definition, we have:

c(p3i+1 )+3i+1DT (p3i+1 )≤c(p3i )+3i+1DT (p3i ),

and

c(p3i+1 )+3iDT (p3i+1 ) ≥ c(p3i )+3iDT (p3i ),

Then

L(3i+1)− L(3i)= c(p3i+1 )+3i+1(D(p3i+1 )− R)T − [c(p3i )+3i(D(p3i )− R)T ]

= c(p3i+1 )+3i(D(p3i+1 )− R)T + (3i+1−3i)

×(D(p3i+1 )− R)T − [c(p3i )+3i(D(p3i )− R)T ]≥ c(p3i )+3i(D(p3i )− R)T + (3i+1

−3i)×(D(p3i+1 )− R)T − [c(p3i )+3i(D(p3i )− R)T ]

= (3i+1−3i)(D(p3i+1 )− R)T

= (λi+1b − λib)(d(p3i+1 )− rb)

≥ |λi+1b − λib|.

Case 2: λi+1b < λib.By Theorem 1 and Lemma 1,

∃p3i+1 and db(p3i+1 ) < rb.

The rest of the proof is similar to Case 1.Hence

|L(3j)− L(31)|= |L(32)− L(31)+ L(33)− L(32)+ · · · + L(3j)− L(3j−1)

|

= |L(32)− L(31)| + |L(33)− L(32)|+ · · · + |L(3j)− L(3j−1)

|

≥ |32−31

|1 + |33−32

|1 + · · · + |3j−3j−1

|1

≥ |3j−31

|1.

The second equality holds because L(31) <

L(32) < · · · < L(3j). �Obviously, if the while-loop in Step 2 in Figure 4 termi-

nates in a finite number of steps, the multiplier is pseudooptimal by definition. If the while loop does not terminate in afinite number of steps (this occurs only when infinite machineprecision is assumed, in practice, GEN-LARAC terminates infinite steps), we have the following theorem.Theorem 5: Let {3i

} be a consecutive sequence ofmultipliers generated in the same while-loop in Step 2in Figure 4. Then the limit point of {L(3i)} is pseudo optimal.

Proof: Since L(31) < L(32) < · · · and {3i} is

bounded from above, limss→∞L(3s) exists and is denotedas L∗. We next show the vector limss→∞3s also exists.By Lemma 6, ∀s, j > 0,

|3s+j−3s|1 ≤ |L(3s+j)− L(3s)|

By Cauchy criterion, limss→∞3s exists. We denoteit as 3∗.We label all the paths in Pst as p1, p2, . . . , pN such

that c3∗ (p1) ≤ c3∗ (p2), . . . , c3∗ (pN ). Obviously p1 is a3∗-minimal path.Let

θ = min {c3∗ (pj)− c3∗ (pi)|∀pi, pj ∈ Pst , c3∗ (pj)

− c3∗ (pi) > 0},

and

π = max {dw(pj)− dw(pi)|∀pi, pj ∈ Pst , w = 1, 2, . . . , k}.





Let M be a large number, such that ∀ t ≥ M ,|3∗ −3t

|1 < θ/(2π ).Consider any component j ∈ {1, 2, . . . , k}, of the

multiplier after computing

arg maxξ≥0L(λ1, . . . , λj−1, ξ, λj+1, . . . , λk )

in iteration t ≥ M .By Theorem 1,

∃ptc and ptd ∈ Pt3 and dj(ptc) ≥ lj ≥ dj(p

td ).

It suffices to show P3t ⊆ P3∗ .Given p3t ∈ P3t , we shall show

c3∗ (p3t ) = c3∗ (p1).

We have

0 ≤ c3t (p1)− c3t (p3t )

= [c(p1)+ d3t (p1)]− [c(p3t )+ d3t (p3t )]

= c(p1)+ d3∗ (p1)− [c(p3t )+ d3∗ (p3t )]

+(3t−3∗)(D(p1)− D(p3t ))T

= c3∗ (p1)− c3∗ (p3t )+ (3t−3∗)(D(p1)

−D(p3t ))T

Since

|(3t−3∗)(D(pi)− D(pj))T | ≤ π |(3t

−3∗)|1 ≤ θ/2,

0 ≤ c3t (p1)− c3t (p3t ) ≤ c3∗ (p1)− c3∗ (p3t )+ θ/2.

Then

c3∗ (p3t )− c3∗ (p1) ≤ θ/2,

which implies that c3∗ (p3t ) = c3∗ (p1).Hence,

∀ j ∈ {1, 2, . . . , k}, ∃p∗c , p∗d ∈ P3∗ , dj(p

∗c ) ≥ lj ≥ dj(p

∗d ).

�

E. SIMULATIONWe use COPT, OPT, and POPT to denote the cost of theoptimal path to the QSR(k) problem, the optimal value,and the pseudo optimal value of the Lagrangian function,respectively. In our simulation, we first verify that the objec-tives at pseudo optimal points are very close to the optimalobjectives. We use 3 types of graphs: Power-law out-degreegraph (PLO) [23], Waxman’s random graph (RAN) [31], andregular graph (REG) [27]. The number of weights chosen are4, 8 and 12, i.e., k = 4, 8, and 12. Link weights are ran-dom even integers uniformly distributed between 2 and 200.To decide proper delay bounds, we randomly choose an s− tpath pran and set ri = (1 + ε)di(pran) where ε is a randomvariable evenly distributed in interval [−0.25, 0.25]. For eachtype of topology, 10 different graphs with valid delay boundsand random source and target nodes are generated. The resultsreported are averaged over the 10 instances.

TABLE 1. Quality of pseudo-optimal paths.

FIGURE 5. Comparision of GEN-LARAC, Hull approach, andsubgradient method.

We use the following two metrics to measure the quality ofpath p in Table 1.

g(p) = c(p)/POPT and f (p) = maxi=1,2,...,kdi(p)/ri,

where g(p) is the upper bound of the gap between the cost ofp and COPT . The f (p) indicates the degree of violation of pto the constraints on its delays.

In Figure 5, the label LARAC-REG means the resultsobtained by running GEN-LARAC algorithm on regulargraphs. Other labels can be interpreted similarly. We onlyreport the results on regular graphs and random graphs forbetter visibility.

We conducted extensive experiments to compare ouralgorithm with the Hull approach [19], the subgradientmethod [1], and the general-purpose LP solver CPLEX.Because the four approaches share the same objective,i.e., maximizing the Lagrangian function, they always obtainsimilar results. We only report the number of shortest pathcomputations which dominate the running time of all thefirst three algorithms. All algorithms terminate when theyhave reached 99% of the OPT. Generally, GEN-LARACalgorithm and Hull approach are faster than the subgradientmethods and CPLEX (See [41] for the comparison of Hullapproach and CPLEX). But GEN-LARAC andHull approachbeat each other on different graphs. Figure 5 shows that on theregular graph, GEN-LARAC is the fastest. But for the randomand Power-law out degree graphs, the Hull approach is thefastest. The probable reason is that the number of s− t pathsis relatively small in these two types of graphs because thelength (number of hops) of s− t paths is short even when thenumber of nodes is large. This will bias the results in favor of




Hull approach which adds one s−t path into the linear systemin each iteration [19]. We choose the regular graph becausewe have a better control of the length of s− t paths.

IV. SUMMARY AND CONCLUSIONIn this paper we developed a new approach to the QoSrouting problem involving multiple additive constraints. Ourapproach uses the LARAC algorithm as a building block andcombines it with certain ideas from mathematical program-ming to design a method that progressively improves thevalue of the Lagrangian function until optimum is reached.The algorithm is analyzed and its convergence property hasbeen established. Simulation results comparing our approachwith two other approaches show that the new approach isquite competitive.

Since the LARAC algorithm is applicable for the generalclass of optimization problems (involving one additive delayconstraint) studied in [3] our approach can also be extendedfor this class of problems whenever an algorithm for theunderlying optimization problem (such as Dijkstra’s algo-rithm for the shortest path problem) is available. In otherwords, besides making an important contribution to the QoSrouting area, this paper contains a significant advance to thegeneral area of combinatorial optimization.

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YING XIAO received the Ph.D. degree incomputer science from the University ofOklahoma, Norman, USA, in 2005. His researchinterests include graph theory, combinatorialoptimization, distributed systems and networks,statistical inference, and machine learning.

KRISHNAIYAN THULASIRAMAN (M’72–SM’84–F’90) received the bachelor’s and master’sdegrees in electrical engineering from the Univer-sity of Madras, in 1963 and 1965, respectively, andthe Ph.D. degree from IIT Madras, India, in 1968.He was on the Faculty of the Electrical Engineer-ing and Computer Science Departments of IITMfrom 1965 to 1981. He was a Professor and theChair of the Electrical and Computer EngineeringDepartment with Concordia University, Montreal,

Canada, from 1981 to 1994 and 1993 to 1994. Since 1994, he has been theHitachi Chair and Professor with the School of Computer Science, Universityof Oklahoma, Norman. He has authored over 100 papers in archival journals.He has co-authored the books entitled Graphs, Networks, and Algorithms(New York: Wiley, 1981) and Graphs: Theory and Algorithms (New York:Wiley, 1992) with M. N. S. Swamy, and several book chapters. He holdsa U.S. patent in distributed quality of service routing. His research interestshave been in graph theory, combinatorial optimization, algorithms and appli-cations in a variety of areas in computer science and electrical engineering,in particular, fault diagnosis, tolerance, and testing. His most recent interestis in the emerging area of network science and engineering. He has receivedseveral awards and honors, such as the Distinguished Alumnus Award ofIIT Madras, the IEEE Circuits and Systems Society Technical AchievementAward, and the IEEE CAS Society Golden Jubilee Medal. He was a SeniorFellow of the Japan Society for Promotion of Science, the GopalakrishnanEndowed Chair Professor of IIT Madras, and a fellow of the AmericanAssociation for the Advancement of Science and the European Academyof Sciences.

GUOLIANG XUE (M’98–SM’99–F’11) receivedthe B.S. degree inmathematics and theM.S. degreein operations research from Qufu NormalUniversity, Qufu, China, in 1981 and 1984, respec-tively, and the Ph.D. degree in computer sciencefrom the University of Minnesota, Minnepolis,MN, USA, in 1991. He is a Professor ofComputer Science and Engineering withArizona State University, Tempe, AZ, USA.His research interests include quality of service

provisioning, resource allocation, survivability, security, and privacy issuesin networks. He received best paper awards at the IEEE Globecom’2007,the IEEE ICC’2011, the IEEE MASS’2011, the IEEE Globecom’2011,and the IEEE ICC’2012, and the Best Paper Runner-Up Award at theIEEE ICNP’2010. He is an Associate Editor of the IEEE TRANSACTIONS ON

NETWORKING, ACM Transactions on Networking, and the IEEE NETWORK,and a member of the Standing Committee of the IEEE INFOCOM.He was the General Co-Chair of the IEEE Conference on Communica-tions and Network Security in 2014. He was a Keynote Speaker at theIEEE LCN’2011, the TCP Co-Chair of the IEEE INFOCOM’2010, and aCo-Organizer of the ARO Workshop on Human Centric Computingwith Collective Intelligence-Challenges and Research Directions in 2012.He served on the Editorial Boards of Computer Networks, the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS, and the IEEE TRANSACTIONS

ON CIRCUITS AND SYSTEMS. He was a Distinguished Lecturer of the IEEECommunications Society.

MAMTA YADAV (S’13) received thebachelor’s degree in computer science fromthe Institute of Engineering and Technology,Rajasthan University, Rajasthan, India, in 2005,and the master’s degree in computer science fromOklahoma City University, OK, USA, in 2008.She is currently pursuing the Ph.D. degreein computer science with the University ofOklahoma, Norman, USA. Her research interestslie in the broad domain of topological study of

complex networks, graph theory, and network science and engineering, inparticular, community detection in complex networks using the topologicaland structural analysis approaches.


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