J. Parallel Distrib. Comput. 68 (2008) 939–948www.elsevier.com/locate/jpdc

Approximating the traffic grooming problem in tree and star networksI

Michele Flamminia, Gianpiero Monacoa, Luca Moscardellia,∗, Mordechai Shalomb, Shmuel Zaksb

a Dipartmento di Informatica, Universita degli Studi dell’Aquila, L’Aquila, Italyb Department of Computer Science, Technion, Haifa, Israel

Received 21 December 2006; received in revised form 9 September 2007; accepted 9 January 2008Available online 4 March 2008

Abstract

We consider the problem of grooming paths in all-optical networks with tree topology so as to minimize the switching cost, measured by thetotal number of used ADMs. We first present efficient approximation algorithms with approximation factor of 2 ln(δ · g) + o(ln(δ · g)) for anyfixed node degree bound δ and grooming factor g, and 2 ln g + o(ln g) in unbounded degree directed trees, respectively. In the attempt to extendour results to general undirected trees, we completely characterize the complexity of the problem in star networks by providing polynomial timeoptimal algorithms for g ≤ 2 and proving the intractability of the problem for any fixed g > 2. While for general topologies, the problem wasknown to be NP-hard g not constant, the complexity for fixed values of g was still an open question.c© 2008 Elsevier Inc. All rights reserved.

Keywords: Optical networks; Wavelength division multiplexing (WDM); Add-drop multiplexer (ADM); Traffic grooming; Tree networks

1. Introduction

All-optical networks have been largely investigated inrecent years, as they promise data transmission rates severalorders of magnitudes higher than current networks [4,10,20,21,23–25]. Major applications are in video conferencing,scientific visualization and real-time medical imaging, high-speed supercomputing and distributed computing [11,20,25].

The key to high speeds in all-optical networks is in maintain-ing the signal in optical form, thereby avoiding the prohibitiveoverhead of conversion to and from the electrical form at theintermediate nodes. The high bandwidth of the optical fiberis utilized through wavelength-division multiplexing: two sig-nals connecting different source–destination pairs may share alink, provided they are transmitted on carriers having differentwavelengths (or colors) of light. Since the optical spectrum is

I This research was partly supported by the EU Project “Graphs andAlgorithms in Communication Networks (GRAAL)” — COST Action TIST293.∗ Corresponding author.

E-mail addresses: flammini@di.univaq.it (M. Flammini),gianpiero.monaco@di.univaq.it (G. Monaco), moscardelli@di.univaq.it(L. Moscardelli), cmshalom@cs.technion.ac.il (M. Shalom),zaks@cs.technion.ac.il (S. Zaks).

0743-7315/$ - see front matter c© 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jpdc.2008.01.003

a scarce resource, communication patterns in different topolo-gies are often designed so as to minimize the total number ofcolors used, and also to allow comparison with the trivial lowerbound provided by maximum load – the maximum number ofconnection paths sharing the same physical edge (see [1,17] fora survey of the main related results).

As the various parameters comprising the switchingmechanism in these networks became clearer, the focus ofstudies shifted. Today a large portion of research concentrateson the total hardware cost. This is modeled by consideringthe basic switching unit of Add-Drop-Multiplexer (ADM),and focusing on the total number of these ADMs. The keypoint here is that each lightpath uses two ADMs, one at eachendpoint. If two adjacent lightpaths are assigned the samewavelength, then they can use the same ADM. An ADM maybe shared by at most two lightpaths.

Moreover, in studying the hardware cost, the issue ofgrooming became central. This problem stems from the factthat the network usually supports traffic at rates which arelower than the full wavelength capacity. Therefore, the networkoperator has to be able to put together (=groom) low-capacitydemands into the high capacity fibers. In graph-theoretic terms,this can be viewed as assigning colors to the lightpaths so thatat most g of them (g being the grooming factor) can share oneedge. In terms of ADMs, each lightpath uses two ADMs, one at

940 M. Flammini et al. / J. Parallel Distrib. Comput. 68 (2008) 939–948

1 This is a generalization of the problem of minimizing the number of ADMs,when traffic grooming is allowed. When restricting to path and ring networks,this means that each of the two edges at a node can have at most g lightpathsof each color; however, when considering a general network, our definitionpermits splitting of groomed lightpaths at a node, which requires a specialtreatment by the optical switch. See discussion at the end of Section 5.

each endpoint, and in case g lightpaths of the same wavelengthenter through the same edge to one node, they can all use thesame ADM (thus saving g−1 ADMs). Moreover, all the same-colored paths ending at the node through two given incidentedges can share the same ADM. The goal is to minimize thetotal number of ADMs.

The problem of minimizing the number of ADMs for thecase g = 1 was introduced in [18] for ring networks. For sucha topology it was shown to be NP-complete in [13] and anapproximation algorithm with approximation ratio of 3/2 waspresented in [6] and improved in [27,14] to 10/7+ ε and 10/7,respectively. For general topologies [13] described an algorithmwith approximation ratio of 8/5 and an improved (3/2 + ε)-approximation algorithm was presented in [5].

The notion of traffic grooming (g > 1) was introducedin [19] for ring topologies. In such a setting, the ADMsminimization problem was shown to be NP-complete in [9]for rings and for general g, i.e., with g being part of theinput instance. Moreover, always for general g, in [12] theauthors show that the grooming problem is NP-complete evenfor paths and star networks. Prior to this paper, the complexityof the problem for fixed values of g and an unbounded numberof wavelength have been known only for path networks:in [26] the problem is shown to be NP-complete, thus implyingthe NP-completeness for trees (even degree-bounded ones ordirected ones). Finally, an algorithm with approximation ratioof 2 ln g for any fixed g in a ring topology has been givenin [15].

The all-to-all traffic case, in which there is the same demandbetween each pair of nodes, has been studied in [3,9] for variousvalues of g, and an optimal construction for the case g = 2 inpath networks has been given in [2].

In a different scenario, the problem of minimizing hardwarecomponents in optical networks – using grooming in orderto exploit large bandwidth – has been studied in [8] for ringnetworks and in [7] for stars networks.

In this paper we extend the results in [15] to trees. Namely,we provide polynomial time algorithms with approximationfactors of 2 ln(∆ · g) + o(ln(∆ · g)) for any fixed node degreebound ∆ and 2 ln g + o(ln g) in unbounded degree directedtrees, e.g. with all the edges oriented from the root to the leavesor vice versa. In the attempt to extend our results to generalundirected trees, we completely characterize the complexity ofthe problem, that is for every fixed value of g, in star networks.Such a topology is of interest in its own right, as it often arisesin practice in the interconnection of LANs or MANs with awide area backbone. In particular we provide polynomial timeoptimal algorithms for g ≤ 2 and prove the intractability of theproblem for any fixed g > 2.

The paper is organized as follows. In the next section, wedescribe the problem and introduce some useful definitions.The algorithms for tree topologies are presented and analyzedin Section 3. In Section 4 we describe the polynomial algorithmfor the star topology for g ≤ 2 and the NP-completeness prooffor g ≥ 3. Finally, in Section 5 we give some concludingremarks and discuss some open questions.

2. Problem definition

An instance of the traffic grooming problem is a triple(G, P, g) where G = (V, E) is a graph, P is a set of simplepaths in G and g is a positive integer, namely the groomingfactor.

A coloring (or wavelength assignment) of (G, P) is afunction w : P 7→ N+ = {1, 2, . . .}. Given any subset Q ⊆ P ,let w(Q) = {w(p)|p ∈ Q} be the set of all the colors assignedby w to the paths in Q. Moreover, for a coloring w, a color λ

and any Q ⊆ P , Qwλ is the subset of paths from Q colored

λ by w. A proper coloring (or wavelength assignment) w of(G, P, g) is a coloring of P in which for any edge e at most gpaths using e are colored with the same color.1

A coloring w is a 1-coloring of Q ⊆ P , if it colors thepaths of Q using one color. A set Q is 1-colorable if thereexists a proper 1-coloring for it. As already remarked in theintroduction, every colored path p ∈ P needs at least one ADMat each of its endpoint nodes. Given a grooming factor g, atmost g paths incident to a node through the same edge can usethe same ADM, and such an ADM can be shared also by the atmost g path incident to the same node through another incidentedge.

Given a 1-colorable subset Q ⊆ P , #ADM1COLOR(Q)

is the minimum number of ADMs needed by the paths in Q(when receiving the same color). More formally, since an ADMoperating at node v can serve at most g requests using anedge incident to v and other g requests using a second edge

incident to v, then #ADM1COLOR(Q) =∑

v∈V

⌈f (v)

2

⌉, where

f (v) is the number of distinct network edges incident to v

used by the paths in Q. For a coloring w of P , #ADMwλ is

the minimum number of ADMs operating at wavelength λ inall the network and #ADMw is the minimum total number ofADMs in the network, that is for all the used wavelengths. Thetraffic grooming problem is the optimization problem of findinga proper coloring w of (G, P, g) minimizing #ADMw.

A possible instance of the traffic grooming problem with anoptimal solution minimizing the number of ADMs is shown inFig. 1.

3. Approximated solutions for tree topologies

In this section we present and analyze the approximationalgorithm for tree topologies. We first give a general descriptionin terms of a generic parameter k, depending only on g, thatwill be properly determined in the analysis, depending on thetree topology (bounded degree trees or directed trees). Wewill always implicitly assume trees rooted at a distinguishedarbitrarily chosen node.

M. Flammini et al. / J. Parallel Distrib. Comput. 68 (2008) 939–948 941

Fig. 1. (a) An optical network having tree topology, with 5 communicationrequests. (b) An optimal solution of the grooming problem for g = 2, using 6ADMs.

3.1. The general algorithm GROOMBYSC(k)

The general algorithm GROOMBYSC(k) described belowhas three phases. During Phase 1 it computes 1-colorable setsand their corresponding weights. It considers subsets of thepaths P of size at most k · g. Whenever a 1-colorable set Qis found, it is added to the list of relevant sets, together with acorresponding weight #ADM1COLOR(Q), that is the minimumnumber of ADMs needed by Q (when its path receive thesame color). In Phase 2 it finds a set cover of P using subsetscalculated in Phase 1. It uses the GREEDYSC approximationalgorithm for the minimum weight set cover problem presentedin [22]. In Phase 3 it transforms the set cover into a partition byeliminating intersections and then colors the paths belonging toeach set with the same color, different from the one of the othersets of the partition.

(1) Phase 1- Prepare the input for GREEDY:S← ∅For each Q ⊆ P , such that |Q| ≤ k · g {

If Q is 1-colorable then {S← S ∪ {Q}weight[Q] = #ADM1COLOR(Q)

}

}

(2) Phase 2- Run GREEDYSC:SC ← GREEDYSC(S, P, weight).Assume w.l.o.g. SC = {S1, S2, . . . , SW }

(3) Phase 3- Transform the Set Cover SC into a PartitionPART:

PART ← ∅For i = 1 to W {PART i ← Si }

As long as there are two intersecting sets PART i ,PART j {

PART i ← PART i \ PART j}

For λ = 1 to W {PART ← PART ∪ {PARTλ}

For each p ∈ PARTλ{w(p) = λ}

}

Algorithm GROOMBYSC(k) is a simple modification of theone presented in [15], where Phase 1 is slightly simplified andthe notion of weighting is extended. In particular, in order toapply the algorithm of [15] for trees, we extended the notion ofweights of the sets to fit to the problem definition (the notionin [15] didn’t apply for trees; it was the number of endpointsof the paths, while for trees we need the minimum number ofADMs needed assuming that all paths receive the same color).

An execution of GROOMBYSC(

32

)for g = 2 on the

instance depicted in Fig. 1 is shown in Table 1 and in Figs. 2and 3. More precisely, each row of Table 1 represents a 1-colorable subset of requests considered in Phase 1 of thealgorithm (with at most k · g = 3 requests), and the subsetscontaining exactly 3 requests are shown in Fig. 2. The greedyapproximation algorithm for the minimum weight set coverproblem presented in [22] builds the set cover solution byconsidering at each step the set Q j having the smallest cost

per request c j , i.e. such that c j =weight[Q j ]

x j= mini

weight[Qi ]xi

,where xi is the number of new requests covered by set Qi . Thesecond and third column of Table 1 show the cost per request ofthe considered sets at the first and second step of GREEDYSC,respectively. The sets with minimum cost per request chosenby the algorithm are highlighted by a •. Since such sets alreadyform a partition of the requests, Phase 3 simply assigns colorsto the requests belonging to them such that all and only therequests in a same set are colored with the same color; in Fig. 3

the output of GROOMBYSC(

32

)is depicted.

The correctness and time complexity of the algorithm followthe arguments of [15]; in particular, it can be shown that itsrunning time is polynomial in n = |P| and m = |E |, for anyfixed k and g and for all instances (G, P, g), by taking intoaccount the fact that #ADM1COLOR(Q) can be calculated inpolynomial time.

As was pointed out in [15], this algorithm is general enoughto be applied to any network topology. Basically, as is oftenthe case in approximation algorithms, it runs in polynomialtime, depending on some constraint; the problem is to prove itsapproximation ratio, and in [15] this is done for ring networks.The novelty and contribution of this paper is in tuning theparameters and, in particular, in extending the analysis sothat it applies for tree topologies. Our analysis for undirectedbounded degree tree networks and for directed tree networksare presented in the following two subsections.

3.2. Bounded degree trees

Now we analyze the algorithm GROOMBYSC(k) applied totrees having their maximum degree equal to ∆.

We will use the shortcut #ADM∗ for #ADMw∗ , and given aset U ⊆ V we will denote by #ADM∗(U ) the number of ADMsused by the optimal solution at nodes belonging to U .

Lemma 1. Given an integer h, there exists a solution SCfor the instance of the set cover problem determined onPhase 2 of GROOMBYSC(k), with k = (∆ − 1)h, such that

weight(SC) ≤(

1+ 2gh

)#ADM∗.

942 M. Flammini et al. / J. Parallel Distrib. Comput. 68 (2008) 939–948

Fig. 2. The sets Q16, . . . , Q24 having cardinality equal to 3 and considered by Phase 1 of the algorithm.

Table 1The 24 sets considered by Phase 1 of the algorithm on the instance depicted in Fig. 1

1-colorable sets Cost per request at step 1 Cost per request at step 2

Q1 = {p1} 2 2Q2 = {p2} 2 ∞

Q3 = {p3} 2 ∞

Q4 = {p4} 2 ∞

Q5 = {p5} 2 2Q6 = {p1, p2}

32 3

Q7 = {p1, p3} 2 4Q8 = {p1, p4} 2 4Q9 = {p1, p5}

32

32 •

Q10 = {p2, p3}32 ∞

Q11 = {p2, p4}32 ∞

Q12 = {p2, p5} 2 4Q13 = {p3, p4}

32 ∞

Q14 = {p3, p5} 2 4Q15 = {p4, p5}

32 3

Q16 = {p1, p2, p3}43 4

Q17 = {p1, p2, p4}43 4

Q18 = {p1, p2, p5}43 2

Q19 = {p1, p3, p4}53 5

Q20 = {p1, p3, p5}53

52

Q21 = {p1, p4, p5}43 2

Q22 = {p2, p3, p4}43 •

Q23 = {p2, p3, p5}53 5

Q24 = {p2, p4, p5}43 4

Proof. Let w∗(P) = {1, 2, . . . , W ∗} and 1 ≤ λ ≤ W ∗.Consider the set V ∗λ of nodes v having at least one ADMoperating at wavelength λ at node v. We divide V ∗λ into sets ofat least h and at most k = (∆−1)h nodes in the following way.We consider a node v in the tree such that the number of nodesbelonging to V ∗λ in the subtree rooted at v is between h and k.Then we remove such a subtree, put in a separate set all thenodes of the subtree belonging to V ∗λ and repeat the procedureon the remaining tree. Clearly, since the maximum degree of thenodes in the tree is ∆, it is always possible to remove a subtreecontaining at least h and at most k nodes of V ∗λ , except for the

last set (in the following, the residual set) that may contain lessthan h nodes.

Let Vλ, j , j = 1, . . . , pλ, be the subsets of nodes obtainedin this way, and qλ the number of them having at least h nodesin V ∗λ . Let rλ < h be the number of nodes in the residual set,if it exists. Notice that if no set with less than h nodes is in thepartition, pλ = qλ and rλ

de f= 0, otherwise pλ = qλ + 1. For

any Vλ, j we define Sλ, j to be the set of paths colored λ by w∗

with both their endpoints in Vλ, j or with one endpoint in Vλ, jand leaving the subtree corresponding to Vλ, j through the edgegoing toward the root. In order to upper bound the number of

M. Flammini et al. / J. Parallel Distrib. Comput. 68 (2008) 939–948 943

Fig. 3. The output of an execution GROOMBYSC(

32

)(for g = 2) on the

instance depicted in Fig. 1. The solution uses 7 ADMs.

such paths, let us charge each of them to one of its endpointnodes reached by the path through the incident link toward theroot. Since by the feasibility of the coloring at most g paths arethen charged to each node in Vλ, j , we have

∣∣Sλ, j∣∣ ≤ g · k.

Notice that every p with w∗(p) = λ is contained in exactly

one set Sλ, j . Therefore SCλde f= ∪ j

{Sλ, j

}is a cover of all the

paths colored λ by w∗. Considering all colors 1 ≤ λ ≤ W ∗ we

conclude that SCde f= ∪

W∗λ=1 SCλ is a cover of P with sets from

S. All that remains is to show that its weight has the claimedproperty.

First observe that #ADM∗λ ≥ hqλ+ rλ. Summing up over allpossible values of λ we obtain #ADM∗ ≥ h

∑λ qλ +

∑λ rλ,

which implies:∑λ

qλ ≤#ADM∗

h. (1)

We claim that for every j ≤ qλ, weight[Sλ, j ] =

#ADM1COLOR(Sλ, j ) ≤ #ADM∗(Vλ, j )+ g. This is because:

• the endpoints of the paths with both endpoints in Sλ, j are inVλ, j ;• the number of paths having only one endpoint in set Vλ, j

is at most g. This follows from the observation that thesepaths should use the unique edge going from the subtreecontaining the nodes in Vλ, j towards the root of the wholetree. As the set Sλ, j is 1-colorable, the number of these pathsis at most g.

For the residual set (which exists only if rλ > 0), we haveweight(Sλ,qλ+1) ≤ #ADM∗(Vλ,qλ+1)+ g · qλ. This is because:

• the endpoints of the paths with both endpoints in Sλ,qλ+1 arein Vλ,qλ+1;• by the same argument as before, the paths having only one

endpoint in Vλ,qλ+1 are at most g in number. When qλ ≥ 1,g ≤ g ·qλ; otherwise qλ = 0 meaning that Vλ,1 is the uniqueset, thus in this case the number of paths having exactly oneendpoint in this set is zero.

Summing up for all 1 ≤ j ≤ qλ + 1 we get weight(SCλ) ≤

#ADM∗λ + 2g · qλ.Summing up for all λ and recalling (1) we get

weight(SC) =∑λ

weight(SCλ)

≤

∑λ

(#ADM∗λ + 2g · qλ)

= #ADM∗ + 2g∑λ

qλ ≤ #ADM∗ + 2g#ADM∗

h

=

(1+

2g

h

)#ADM∗. �

Now, by choosing h = g ln g, since the greedy algorithmfor the Minimum Weight Set Cover Problem is a H f -approximation algorithm, where f is the maximum cardinalityof the sets in the input and H f = 1 + 1

2 + · · · +1f the f -th

harmonic number, it is possible to show the following theorem.

Theorem 2. There is a 2 ln g + ln ∆ + o(ln(∆ · g))-approximation algorithm for the traffic grooming problem inbounded degree tree networks.

Proof. Let Hn = 1+ 12 + · · ·+

1n be the n-th harmonic number

and k = (δ − 1)h, where h will be determined in the followingand δ is the maximum degree of the nodes in the tree. We firstshow that the approximation ratio ρ of GROOMBYSC(k) is at

most Hg·δ·h

(1+ 2g

h

)for the traffic grooming problem in tree

networks with degree bounded by δ.Recall that in the Minimum Weight Set Cover problem each

subset Si has an associated weight weight[Si ]. The weight of acover is the sum of the individual weights of its sets.

Let w and PART1, . . . , PARTW be the coloring and thepartition returned by GROOMBYSC(k), respectively, and w∗ bean optimal coloring.

On the one hand,

#ADMw=

∑λ

#ADMwλ =

∑λ

#ADM1COLOR(PARTλ)

≤

∑λ

#ADM1COLOR(Sλ)

=

∑λ

weight[Sλ] = weight(SC). (2)

On the other hand, GREEDYSC is an H f -approximationalgorithm, where f is the maximum cardinality of the sets inthe input, that is f = g · k. In other words if SC∗ is a minimumweight set cover on the set S, we have

weight(SC) ≤ Hg·k weight(SC∗). (3)

Clearly if SC is an arbitrary set cover of S, by definition

weight(SC∗) ≤ weight(SC). (4)

Combining the inequalities (2)–(4) we get

#ADMw≤ Hg·k weight(SC)

for any set cover SC of S.By Lemma 1, #ADMw

≤ Hg·k

(1+ 2g

h

)#ADM∗ and

therefore ρ = Hg·k

(1+ 2g

h

).

We obtain the claim by substituting h = g ln g:

ρ ≤ Hg2δ ln g

(1+

2ln g

)≤ (1+ ln(g2δ ln g))

(1+

2ln g

)= (1+ 2 ln g + ln δ + ln ln g)

(1+

2ln g

)= 2 ln g + ln δ + o(ln g). �

944 M. Flammini et al. / J. Parallel Distrib. Comput. 68 (2008) 939–948

3.3. Directed trees

We now analyze the algorithm GROOMBYSC(k) applied todirected trees, i.e. to trees having all the edges directed fromthe root towards the leaves or vice versa. Notice that the resultfor this topology is independent of the degree ∆. This stemsfrom the fact that in directed trees lightpaths cannot cross fromone subtree to another (in contrast with the case of undirectedones).

By the same arguments used in the proof of Theorem 2, itis possible to show that GROOMBYSC(k) is a Hg·k(1 +

2gh )

approximation algorithm for the traffic grooming problem indirected tree networks, where k = 3h and h is again equal tog ln g. In the following, we underline the differences betweenthe analyses on the two tree topologies, using the same notationintroduced for bounded degree trees.

The main difference is in the proof of the existence of a set

cover of weight at most(

1+ 2gh

)#ADM∗. Since the maximum

degree of the tree is not bounded, we cannot partition V ∗λ in thesame way using node disjoint subtrees. Thus, we divide V ∗λ intosets of at least h and at most 3h nodes in the following way. Ateach step, we consider a node x such that all its subtrees haveless than h nodes in V ∗λ and there exists a subset of subtreeshaving in total at least h and at most 3h nodes belonging toV ∗λ satisfying the following property: in the optimal solution noADM operating with color λ at node x is shared by requestsbelonging to different components. Notice that this is alwayspossible since without loss of generality we can restrict tooptimal solutions having only one ADM at node x with colorλ for all the requests coming from each single subtree of x andsince by coupling subtrees with requests sharing the same ADMat node x we obtain components having a number of nodes notexceeding 3h (if a subset has less than h nodes, adding twosubtrees sharing an ADM at x in the optimal solution will yieldat most 3h nodes). Thus we remove such subtrees inserting alltheir nodes and eventually node x in a single component; moreprecisely, node x is inserted in the component if and only ifthere exists an ADM operating at color λ in the optimal solutionat x used by requests belonging to a subtree of the componentand other ones crossing the edge going toward the root. Finally,x is removed only if it has not further sons and has been insertedin one component. We keep repeating the same step until anode x with the same property is present in the remaining tree.Clearly, since we remove node x only if it does not disconnectthe tree, it is always possible to select a set containing at leasth and at most 3h nodes of V ∗λ , except for the last (residual) setthat may contain less than h nodes.

Since the tree is directed, no request can step throughtwo subtrees of a same parent node, and thus through twocomponents composed by subtrees having the same parentnode. Therefore, we can again claim that weight(SCλ) ≤

#ADM∗λ + 2g · qλ, by conveniently partitioning the ADMsof the optimal solution with special regards for the ADMs atthe parent nodes of the subtrees belonging to the components;more precisely, we partition the ADMs at such nodes so that anADM shared in the optimal solution by requests belonging to a

component is counted in the set of the partition correspondingto such a component.

Therefore, it is possible to prove the following theorem.

Theorem 3. There is a 2 ln g + o(ln g)-approximation algo-rithm for the traffic grooming problem in directed tree networks.

Proof. The approximation ratio ρ of GROOMBYSC(k) is at

most Hg·3h

(1+ 2g

h

).

We substitute h = g ln g and get:

ρ ≤ H3g2 ln g

(1+

2ln g

)≤ (1+ ln(3g2 ln g))

(1+

2ln g

)= (1+ ln 3+ 2 ln g + ln ln g)

(1+

2ln g

)= 2 ln g + o(ln g). �

4. Star topology

In this section, we consider star topologies having one hubnode (node 0) of degree n and n nodes 1, . . . , n of degree 1,connected to the hub node. We first present and analyze optimalalgorithms for g ≤ 2 and then show that the problem is NP-complete for every g ≥ 3.

4.1. Optimal algorithms for g ≤ 2

The case g = 1 can be trivially solved by arbitrarily pairingall the possible requests having the hub node as an endpoint(no saving is possible for the other paths). Thus, we restrict tothe case g = 2 considering instances of the form (G, P, 2)

with G = (V, E), where V = {0, . . . , n} and E = {{0, i}|i =1, . . . , n}.

4.1.1. Algorithm GROOMSTARThe algorithm has three phases. During Phase 1 it creates

a new graph H having the same node set of G and an edgefor any request in P between its endpoints. In Phase 2 itfinds and removes all the cycles in H and colors the requestscorresponding to their edges assigning a different wavelengthto each cycle. Finally, in Phase 3 it finds all the remaining pathsbetween nodes having an odd degree in H , and properly colorsthe corresponding requests assigning a different wavelength toeach path.

(1) Phase 1- Create graph H :E ← ∅For each p ∈ P , such that p is between i and j

E ← E ∪ {{i, j}}H ← (V, E)

(2) Phase 2- Find and color all cycles in H :λ← 1While there exists a cycle C = {e1, . . . , ek} in H {

E ← E \ CH ← (V, E)

For h = 1 to k {Let w(p) = λ where p ∈ P is the pathcorresponding to eh

M. Flammini et al. / J. Parallel Distrib. Comput. 68 (2008) 939–948 945

}

λ← λ+ 1}

(3) Phase 3- Color remaining paths in H :While there exist x, y ∈ V connected and having odddegree with respect to edges in E {

Let C = {e1, . . . , ek} be the sequence of edgesconnecting x to yE ← E \ CH ← (V, E)

For h = 1 to k {Let w(p) = λ where p ∈ P is the pathcorresponding to eh

}

λ← λ+ 1}

4.1.2. Running timeThe running time of the algorithm is clearly a polynomial.

In fact, Phase 1 has a time complexity linear in the size ofthe instance; in Phase 2 and Phase 3 the problem of findinga cycle or a sequence of edges connecting two nodes is solvablein linear time by visiting the graph. These problems have to becomputed at most |P| times.

4.1.3. Correctness and optimalityLet xi , i = 1, . . . , n, be the number of paths of two edges in

P having an endpoint in node i ; similarly, let yi be the numberof paths of one edge in P having an endpoint in node i .

Lemma 4. The number of ADMs needed for the traffic

grooming problem in a star network is at least⌈∑n

i=1 yi2

⌉+∑n

i=1

⌈ xi+yi2

⌉.

Proof. For each node 1, . . . , n we need at least one ADM foreach pair of paths having an endpoint in it, i.e. a total of

⌈ xi+yi2

⌉ADMs. For the hub node 0, since for each edge there exists atmost one path p ∈ P crossing it and having node 0 as endpoint,we again need at least one ADM for each pair of such paths,

i.e. a total of⌈∑n

i=1 yi2

⌉ADMs. �

Theorem 5. Algorithm GROOMSTAR returns an optimal

solution using exactly⌈∑n

i=1 yi2

⌉+

∑ni=1

⌈ xi+yi2

⌉ADMs.

Proof. During Phase 2, cycles are found and colored. Noticethat in each cycle we use one ADM for each pair of paths andeliminating it the parity of the degree of nodes in H is notmodified.

In Phase 3, at each step the number of nodes having anodd degree is decreased by 2. Since there are no cycles at thebeginning of this phase, it is always possible to find a sequenceof edges starting and ending at two nodes with odd degrees andto color the corresponding paths with the same color. Noticethat at every node we use one ADM for each pair of paths,except for the nodes having an odd degree. Since such nodes

after removing the sequence of edges from H (correspondingto a path between nodes of odd degree) become of even degree,only once such a situation occurs at a node having an initialodd degree. Thus, the claim directly follows by Lemma 4 andby observing that at each node we use a number of ADMs equalto the number of possible pairings of incident paths, plus one incase of odd degree. �

4.2. NP-completeness for g > 2

Now we turn to show that determining a solution using aminimum number of ADMs for any g > 2 is an intractableproblem. Before proving it formally, we need to show the NP-completeness of the following decision problem.Edge Partition in δ-regular graphsINPUT: undirected graph G = (V, E).QUESTION: is it possible to partition the edge set E in ksubsets E1, . . . , Ek for some k > 0, each inducing a δ-regularsubgraph Gl = (Vl , El), 1 ≤ l ≤ k, that is with each nodehaving degree exactly δ?

Lemma 6. For every δ ≥ 3 Edge Partition in δ-regular graphsis an NP-complete problem.

Proof. For the sake of simplicity, we first prove the claim forδ = 3. In order to show the NP-completeness, we provide apolynomial time transformation from the Exact Cover by 3-Setsproblem (known to be NP-complete; see [16]). In this problemwe have a universe set U = {o1, . . . , om} of m elements withm multiple of 3, a family S = {S1, . . . , Sh} of h subsets of U ,each containing exactly 3 elements, and we want to decide ifthere exist m/3 subsets S j1 , . . . , S jm/3 that partition U , i.e. such

that⋃m/3

i=1 S ji = U .Starting from an instance of Exact Cover by 3-Sets, we

construct a graph G = (V, E) whose edges can be partitionedin δ-regular graphs if and only such an instance admits apartition.

For any o j ∈ U , let n j be the number of subsets in Scontaining o j . Then G = (V, E), where V = V1∪V2∪V3∪V4and E = E1 ∪ E2 ∪ E3 ∪ E4 ∪ E5, (see Fig. 4) with:

V1 = {ui | i = 1, . . . , h},

V2 = {v j | j = 1, . . . , m},

V3 = {w1, j,l , w2, j,l | j = 1, . . . , m, l = 1, . . . , n j },

V4 = {z1, j,l , z2, j,l | j = 1, . . . , m, l = 1, . . . , n j }, and

E1 = {{ui , v j } | o j ∈ Si },

E2 = {{v j , w1, j,l}, {v j , w2, j,l} | j = 1, . . . , m, l = 1, . . . , n j },

E3 = {{w1, j,l , z1, j,l}, {w1, j,l , z2, j,l}, {w2, j,l , z1, j,l},

{w2, j,l , z2, j,l} | j = 1, . . . , m, l = 1, . . . , n j },

E4 = {{z1, j,l , z2, j,l} | j = 1, . . . , m, l = 2, . . . , n j },

E5 = {{z2, j,1, z1, j+1,1} | j = 1, . . . , m − 1, l = 2, . . . , n j }

∪{{z2,m,1, z1,1,1}}.

Informally, in the reduction graph each subset Si ∈ Scorresponds to the subgraph induced by vertex ui and thevertices v j such that o j ∈ Si , which are all connected to ui .

946 M. Flammini et al. / J. Parallel Distrib. Comput. 68 (2008) 939–948

Fig. 4. The reduction graph for Edge Partition in 3-regular graphs.

Fec

Taatl

tp

ract

Moreover, at each o j , for each Si containing it there is anattached auxiliary component induced by v j and vertices w1, j,l ,w2, j,l , z1, j,l and z2, j,l in V3 ∪ V4 connected among themselvesby the edges in E3 ∪ E4 so that they all have degree exactly 3,except v j . Notice that the first auxiliary component of each v j isslightly different from the other ones, as z1, j,l and z2, j,l are notadjacent. In fact, the first auxiliary component of v j for j < mis connected to the first of v j+1 by an edge in E5, and the firstof vm with the one of v1, still maintaining degree 3 for all nodesw1, j,1, w2, j,1, z1, j,1 and z2, j,1. The key property here is that,since for a node of degree 3 either all the incident edges mustbe in a partition inducing a 3-regular graph or none of them,in any partition all the edges of the first auxiliary componentsmust belong to the same partition set.

Assume that the instance of Exact Cover by 3-Sets admitsthe required partition, and without loss of generality letS1, . . . , Sm/3 be the partition subsets. Then G can be edgepartitioned into 3-regular graphs as follows. In the first partitionwe insert the edges of all the first auxiliary components,including the one connective two consecutive ones, plus theedges incident vertices u1, . . . , um/3. In this way, since everyui ∈ V1 has degree 3, all the nodes in the auxiliary componentshave degree 3 and each v j has 2 incident edges belongingto its first auxiliary component plus exactly one connectingit to u1, . . . , um/3, all such induced graphs are 3-regular. Theremaining 3-regular components are formed including, for eachui ∈ V1 with i > m/3, the edges incident to ui plus the one of anot yet considered auxiliary component of each of the verticesv such that o j ∈ Si (notice that for each v j they are exactlythe ones needed to be joined to the edges incident to verticesui ∈ V1 with i > m/3).

On the other hand, if G can be edge partitioned into 3-regular graphs, a partition of the universe U can be determinedas follows. Consider the 3-regular subgraph containing allthe first auxiliary components of the nodes v j ∈ V2 (aspreviously mentioned they must be all together). Since eachv j ∈ V2 has exactly two edges belonging to its first auxiliarycomponent, exactly one edge incident to it must be insidethe same subgraph. Such an edge cannot belong to anotherauxiliary component of v j , as all such edges must be in the samesubgraph and this would cause a degree greater than 3 at v j .

ig. 5. The auxiliary components for (a) δ = 5 (the construction can be easilyxtended for any δ odd at least equal to 5), (b) δ = 4, (c) δ = 6 (the constructionan be easily extended for any δ even at least equal to 6).

hus the additional edge of v j must have as the other endpointnode ui ∈ V1. Since this holds for all the nodes v j ∈ V2

nd for every ui ∈ V1 either all its incident edges belong tohe subgraph or none of them, vertices ui ∈ V1 connected to ateast one vertex v j ∈ V2 are connected to exactly 3 vertices ofV2. Since each v j ∈ V2 is connected to exactly one ui ∈ V1,he subsets Si corresponding to such nodes ui ∈ V1 then form aartition of U , hence the claim.

We now show how to extend the proof to every degree δ > 3.Let us first consider the case in which δ is odd. The

eduction is from Exact Cover by δ-Sets. Set vertices ui ∈ V1re connected to element vertices v∈V2 accordingly. Auxiliaryomponents are extended so as to transpose the same argumentso δ-regular graphs in the following way (see Fig. 5):

V3 = {wa, j,l |a = 1, . . . , δ − 1, j = 1, . . . , m, l = 1, . . . , n j },

M. Flammini et al. / J. Parallel Distrib. Comput. 68 (2008) 939–948 947

V4 = {za, j,l |a = 1, . . . , δ − 1, j = 1, . . . , m, l = 1, . . . , n j },

and

E2 = {{v j , wa, j,l} | a = 1, . . . , δ − 1, j = 1, . . . , m,

l = 1, . . . , n j },

E3 = {{wa, j,l , zb, j,l} | a = 1, . . . , δ − 1, b = 1, . . . , δ − 1,

j = 1, . . . , m, l = 1, . . . , n j },

E4 = {{z2a−1, j,l , z2a, j,l}, | a = 1, . . . , (δ − 1)/2,

j = 1, . . . , m, l = 2, . . . , n j },

E5 = {{z2a, j,l , z2a+1, j,l}, | a = 1, . . . , (δ − 3)/2,

j = 1, . . . , m, l = 2, . . . , n j } ∪ {{zδ−1, j,1, z1, j+1,1}, |

j = 1, . . . , m − 1, l = 2, . . . , n j } ∪ {{zδ−1,m,1, z1,1,1}}.

The proof, then, proceeds in a completely analogous way,always exploiting the property that all the edges incident to avertex of degree δ must all be in the same partition set.

It remains to give the proof for the even values of δ ≥ 4.For δ = 4 the reduction is modified in such a way that to

each subset Si ∈ S we now associate two vertices u1,i , u2,i andto each element two vertices v1, j , v2, j . Each u1,i (resp. u2,i ),1 ≤ i ≤ h, is connected to all the v1, j (resp. v2, j ) such thato j ∈ Si . The auxiliary components are shown in Fig. 5. Each isattached to both v1, j and v2, j . Each auxiliary component needsto be in the same partition set, the first auxiliary componentsmust be all together in the same set, in each set an auxiliarycomponent attached to v1, j , v2, j needs to be associated with anedge in E1 incident to v1, j and one incident to v2, j , and finallyas it can be easily checked all the edges incident to verticesu1,i , u2,i must all be in the same set. The proof then follows bythe same arguments for δ = 3.

For δ ≥ 6 the reduction is from Exact Cover by (δ − 1)-Sets. Here each subset Si ∈ S is associated with two verticesu1,i , u2,i connected by an edge and connected to all the verticesv j such that o j ∈ Si . The auxiliary components attachedto vertices v j 1 ≤ j ≤ m, are depicted in Fig. 5. Eachauxiliary component needs to be in the same partition set, thefirst auxiliary components must be all together in the same set,in each set an auxiliary component needs to be associated withtwo edges in E1 incident to v j , and finally all the edges incidentto vertices u1,i , u2,i must all be in the same set. By the sameconsiderations the claim follows. �

We can now show the hardness the grooming problem in starnetworks for every fixed g > 2.

Theorem 7. For any given g > 2, given in input a set ofcommunication requests in a star network, deciding if thereexists a solution of the traffic grooming problem using at mosta given number h of ADMs is NP-complete.

Proof. We prove the claim by reduction from Edge Partitionin δ-regular graphs with δ = g, shown to be NP-complete inthe previous lemma. Given an instance G = (V, E) of such aproblem with V = {1, . . . , n}, we construct in polynomial timean instance of the grooming problem such that there exists asolution using at most 2|E |/g ADMs if and only if the edgesof G can be partitioned in δ-regular graphs. The star networks

has as node set V ∪ {0} and as set of communication pathsP = {〈i, 0, j〉|{i, j} ∈ E}.

Assume that E can be edge-partitioned for some k > 0in k subsets E1, . . . , Ek inducing g-regular subgraphs Gλ =

(Vλ, Eλ), 1 ≤ λ ≤ k, each with a number of nodes |Vλ| =

2|Eλ|/g.Since by the star topology each subset of paths Qλ =

{〈i, 0, j〉|{i, j} ∈ Eλ} is 1-colorable, by letting w(p) = λ foreach p ∈ Qλ, the number of ADMs sufficient for Qλ is |Vλ|.Therefore, there exists a solution using an overall number ofADMs equal to |V1| + · · · + |Vk | = 2(|E1| + · · · + |Ek |)/g =2|E |/g.

Let us assume that E cannot be partitioned so as to induceonly g-regular subgraphs and consider any solution of thegrooming problem using k colors for the paths in P for somek > 0. Let Pλ ⊆ P , 1 ≤ λ ≤ k, the subset of the pathsof P with color λ. Then P1, . . . , Pk induce in a natural waya partition E1, . . . , Ek of E with corresponding subgraphsG1 = (V1, E1), . . . , Gk = (Vk, Ek), each having maximumdegree upper bounded by g, and thus with a number of nodes|Vλ| ≥ 2|Eλ|/g. Moreover, since by hypothesis at least onesubgraph is not g-regular, there must exist l, 1 ≤ λ ≤ k, suchthat |Vλ| > 2|Eλ|/g. Clearly, the number of ADMs used foreach color λ, is at least |Vλ|, so that the overall number ofADMs used by the solution is at least |V1| + · · · + |Vk | >

2(|E1| + · · · + |Ek |)/g = 2|E |/g, thus proving the claim. �

5. Discussion and open problems

In this paper we addressed the traffic grooming problemwith respect to the minimization of ADMs. We presentedapproximation algorithms for tree networks having boundeddegree or directed, with approximation ratios logarithmic in g.Moreover, we provided an optimal polynomial algorithm forthe star topology and grooming factor 2 and showed that theproblem is NP-complete even for such topologies and everyfixed grooming factor at least equal to 3.

The main open problem arising from this paper is thedetermination of an approximation algorithm for general trees.To the best of our knowledge, no approximation algorithmhas been proposed – not even for stars and grooming factor3. Another interesting research direction is the minimizationof ADMs in the setting in which only a limited number ofwavelengths is available.

An additional extension concerns the notion of trafficgrooming. While grooming in path and ring networks isclear, its extension to a general network can be understoodin more than one way. In fact according to our definition,it is possible that two lightpaths with the same wavelengthwill be directed to different endpoints at an intermediatenode having degree at least 3. However, coming back to thehardware switches, one can argue that this splitting mightrequire additional hardware equipment, thus resulting in costfunctions considering not only the number of ADMs butalso these additional components. Investigating the possibleinterpretations of grooming, according to the various hardwarecomponents, calls for more future research.

948 M. Flammini et al. / J. Parallel Distrib. Comput. 68 (2008) 939–948

Acknowledgment

We would like to thank David Coudert for helpful commentsregarding an earlier version of this paper.

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Michele Flammini received the degree in ComputerScience at the University of L’Aquila in 1990 and thePh.D. degree in Computer Science at the Universityof Rome “La Sapienza” in 1995. He is full professorat the Computer Science Department of the Univer-sity of L’Aquila since march 2005. His research inter-ests include algorithms and computational complexity,game theory, communication problems in interconnec-tion networks and routing. He has authored and co-authored more than 90 papers in his fields of interest

published in the most reputed international conferences and journals.

Gianpiero Monaco received the degree in ComputerScience at the University of L’Aquila in 2006. Heis currently a Ph.D. student in Computer Scienceat the University of L’Aquila. His current researchinterests include optimization problems in social andcommunication networks and algorithmic game theory.

Luca Moscardelli received the degree in ComputerScience at the University of L’Aquila in 2004. Heis currently a Ph.D. student in Computer Scienceat the University of L’Aquila. His current researchinterests include optimization problems in social andcommunication networks, and the analysis of theinteraction between selfish agents in non-cooperativesystems.

Mordechai Shalom, holds a B.A. degree in Architec-ture from ITU, Technical University of Istanbul, fromwhich he graduated at 1981. He completed his studiestowards M.Sc. degree at 1986 in the Faculty of Com-puter Science, Technion, from which he received hisPh.D. degree at 2006. He is currently a Teaching Assis-tant at the Technion, and an independent Software Con-sultant. His research interests are Optimization Prob-lems in Optical Networks, Approximation Algorithmsfor Graph Theoretic problems.

Shmuel Zaks received his B.Sc. (cum laude) and M.Sc.degrees in Mathematics from the Technion, Haifa,Israel, in 1971 and 1972, respectively, and his Ph.D.degree in Computer Science from the University ofIllinois at Urbana-Champaign in 1979. He is a fullprofessor at the Department of Computer Science atthe Technion, where he has been since 1979. He isan author over 100 journal and conference papers,which span his research interests, including DistributedComputing, ATM networks, Optical Networks,Graph

and Combinatorial Algorithms, and Discrete Mathematics.