IEEE COMMUNICATIONS LETTERS, VOL. 8, NO. 7, JULY 2004 473

The Most Congested Cutset: Deriving a TightLower Bound for the Chromatic Number in

the RWA ProblemAhmad R. Sharafat, Senior Member, IEEE, and Omid R. Ma’rouzi, Member, IEEE

Abstract—Routing and wavelength assignment (RWA) is an im-portant issue in the design of routing protocols in an all-opticalnetwork. A solution for the RWA problem should determine theleast possible number of wavelengths, also known as the chromaticnumber. In this letter, we introduce the concept of cutset conges-tion, and show that the congestion of the most congested cutset ofa graph representing the network is a tight lower bound for thechromatic number in the RWA problem. In contrast to other ex-isting techniques, our method can be applied to any network andany traffic pattern.

Index Terms—All-optical networks, routing and wavelength as-signment (RWA), chromatic number, cutset congestion.

I. INTRODUCTION

ALL-OPTICAL networks use end-to-end transparentchannels (lightpaths) to connect all source-destination

pairs. A lightpath is a physical directed path (dipath) togetherwith a wavelength assigned to it. In any given node, incominglightpaths with the same wavelength must be routed to differentoutput ports, and lightpaths on a unidirectional optical linkmust have different wavelengths (wavelength interference con-straint). Furthermore, a lightpath must preserve its wavelengthover the entire length of the dipath from source to destination(wavelength continuity constraint).

We model an all-optical network as a symmetric di-rected graph (digraph) , where each arc representsa point-to-point unidirectional fiber optic link. A connectionrequest is an ordered pair of nodes , and an instanceis a multiset of connection requests. A routing is a set ofdirected paths (dipaths), which specifies a physicalpath for connecting all connection requests in an instance ,i.e., . A wavelength assignmentis a function that assigns a set of wavelengths todifferent dipath of , i.e., .

The problem is to find the minimum number ofwavelengths assigned to different routes for connecting all con-nection requests in over topological constraints imposed bydigraph , so that both interference and wavelength continuityconstraints are satisfied. We seek the least possible numberof wavelengths, also known as the chromatic number of theproblem .

Manuscript received August 19, 2003. The associate editor coordinating thereview of this letter and approving it for publication was Prof. J. Evans.

The authors are with the Department of Electrical Engineering, TarbiatModarres University, Tehran, Iran (e-mail: sharafat@isc.iranet.net).

Digital Object Identifier 10.1109/LCOMM.2004.832765

The pioneering work of Pankaj [1] resulted in finding anapproximate lower bound for the chromatic number of the per-mutation routing problem on specific topologies such as shuffleexchange, De Bruijn, and hypercubic networks. However, thesereflect topologies of optical computers, and not of WANs.Aggarwal et al. [2] considered oblivious wavelength routingschemes and derived the lower bound for the chromatic numberfor a permutation traffic pattern on a general network topology.Bermond et al. [3] focused on bounded-degree networks andderived the upper and the lower bounds for one-to-all andall-to-all traffic patterns using the notion of network expansion.A conventional method for estimating the lower bound ofchromatic number for any network topology and any trafficpattern is LP-relaxation [4]. Qiao and Zhang [5] obtained alower bound for the chromatic number in WDM rings. Baroniand Bayvel [6] utilized the notion of limiting cutset to obtaina lower bound for the number of wavelengths in an all-to-alltraffic pattern.

We generalize the notion of limiting cutset and introduce theconcept of the most congested cutset to derive an accurate lowerbound (as compared to approximate values derived in [1] and[2]) for general traffic patterns on any given arbitrary networktopology (in contrast to specific traffic patterns and topologiesin [3], [5], and [6]). We show that the congestion of the mostcongested cutset in digraph is equivalent to a tight lower boundfor the chromatic number of the graph representing the network,and apply an algorithm we developed in [7] to find this lowerbound for some sample networks.

II. THE MOST CONGESTED CUTSET

In this section we introduce some basic concepts.Definition 1—Arc Congestion: For a given arc in a digraphand a routing for a connection requests instance in ,

the number of dipaths that pass through is the arc congestion,denoted by .

Observation 1: The number of assigned wavelengths to allpaths that pass through an arc is the arc congestion. This is adirect consequence of wavelength interference constraint.

Definition 2—Congestion of a Digraph: The congestion of adigraph for a connection requests instance and a routing

is the maximum congestion of all arcs in the digraph, i.e.,.

Observation 2: For a given digraph and a connection re-quests instance , the number of total wavelengths assigned toall paths in is greater than or equal to the congestion of .

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474 IEEE COMMUNICATIONS LETTERS, VOL. 8, NO. 7, JULY 2004

This is a result of applying Observation 1 to the most congestedarc in the graph.

Definition 3—Load of a Digraph: The load of for a con-nection requests instance is the minimum congestion of inany routing for , i.e., .

1) Routing Problem Statement: Given a network as repre-sented by a digraph and a connection requests instance , wewish to find a routing that minimizes the congestion in thedigraph, which is equal to the load of digraph .

Definition 4—Coloring Number: Given a digraph and arouting for a connection requests instance , the coloringnumber, i.e., , is defined as the minimum possiblenumber of wavelengths assigned to all paths of , so that notwo paths with the same wavelength pass through a commonarc and the wavelength continuity constraint is also satisfied.

2) Wavelength Assignment Problem Statement: Given anall-optical network, as represented by a digraph and arouting set for a connection requests instance , we wish tofind the wavelength assignment function under wavelengthinterference and continuity constraints, so that the number ofwavelengths is equal to the coloring number .

Observation 3: .Proof: See [8].

Definition 5—The Chromatic Number of Digraph: Thechromatic number of for a connection requests instance isthe least coloring number in any possible routing for , i.e.,

.3) RWA Problem Statement: Given an all-optical network

as represented by digraph as well as a connection requestsinstance , we wish to find a routing and a wavelength as-signment under wavelength interference and continuity con-straints so that the number of wavelengths is equal to the chro-matic number .

Observation 4: .Definition 6—Minimal Cutset: For a given digraph ,

a subset of arcs is a minimal cutset if and only if deletingall arcs in would divide into two separate, connected, anddirected subgraphs. An isolated node is considered as a con-nected directed subgraph. Each minimal cutset divides the ver-tices of into two disjoint subsets and . We denote a cutsetas . For convenience, we use “cutset” instead of minimalcutset throughout this paper.

Definition 7—Transit Traffic of a Cutset: For a cutsetin a digraph , the transit traffic (or ) is de-fined as the set of connection requests in the connection re-quests instance such that the source vertex is in the sub-graph (or ), and the destination vertex is in subgraph(or ), i.e., (or

).Observation 5: Transit traffic of a cutset is independent of

routing, and depends only on the topology of the cutset and itsconnection requests instance . This is a result of Definitions 6and 7.

Definition 8—Transit Traffic Intensity of a Cutset: Thetransit traffic intensity of a cutset is the ratio of the number ofits transit traffic to the number of arcs (or degree) of cutset,i.e.,

.

Definition 9—Congestion of a Cutset: The least integergreater than or equal to the maximum value of transit trafficintensity of a given cutset in both directions is the congestionof that cutset, i.e.,

.Observation 6: Congestion of a cutset is independent of

routing and only depends on topology of the cutset and itsconnection requests instance . This is a result of Observation5 and Definitions 8 and 9. While congestion of a cutset isindependent of routing, congestion of every arc in digraph(including cutset arcs) depends on the routing. The congestionof a cutset is not the same as the congestion of its arcs.

Theorem 1: Congestion of a cutset is a lower bound for thenumber of wavelengths assigned to transit traffic in that cutset.

Proof: It is shown in [9] that if the average of some inte-gers is greater than the integer , then at least one of theseintegers is greater than or equal to . Without loss of gener-ality, we assume that , so wehave . From Definition 8 weknow that is the average number of transit trafficconnection requests. Since for every connection request thereis one path, then is the average number of pathsthat pass through the arcs of cutset , or equally, is theaverage of arc congestion for the cutset arcs. It is evident that

, and therefore, the con-gestion of at least one of the cutset arcs is greater than or equalto or . Now, from Observation 2, thecongestion of a cutset is a lower bound for the congestion of allarcs in the cutset, or equally, is a lower bound for the number ofwavelengths assigned to the transit traffic that pass through thatcutset.

Definition 10—The Most Congested Cutset (MCCS): For agiven digraph and a connection requests instance , the mostcongested cutset is

for every possible . The congestion ofthe most congested cutset is . It ispossible to have more than one most congested cutset for agiven pair of digraph and connection requests instance.

Theorem 2: The congestion of the most congested cutset of agiven digraph for a connection requests instance is a lowerbound for the load of digraph , i.e., .

Proof: From Theorem 1, the congestion of at least one arcin is greater than or equal to . From Observation6, this congestion is independent of routing. Thus, independentof the routing, the congestion of at least one arc in is notless than . Therefore, noting the statement of the routingproblem , the load of , i.e., can not be less than

.Theorem 3: The congestion of the most congested cutset of a

given digraph for a connection requests instance is a lowerbound for the chromatic number of that digraph, .

Proof: From Theorem 3, . From Ob-servation 4, . Thus, .

III. HOW TO FIND MCCS

To find the most congested cutset for a given problem ,we need to scan all cutsets of the network and calculate the

SHARAFAT AND MA’ROUZI: DERIVING A TIGHT LOWER BOUND FOR THE CHROMATIC NUMBER IN THE RWA PROBLEM 475

congestion for each and every cutset. However, this problem isNP-hard. The conventional approach for scanning all cutsets ofa given graph uses state-space enumeration. This is impracticalbecause its size grows exponentially with the number of nodes.However, it can be improved by utilizing the topological prop-erties of the graph to substantially reduce the size of state-space.Using this concept, Tsukiyama et al. [10] proposed an algorithmfor finding the -mincuts of a given graph in linear timeper cutset. This recursive algorithm enumerates all connected

separating cutsets by considering only certain extensions(1-point extensions) of the separating cutsets. In [7] we modifythe method proposed by Tsukiyama et al. [10] by using iterativecontraction [11] and BFS ordering of vertices to develop a novelrecursive contraction algorithm for scanning (enumerating andlisting) all minimal cutsets (in contrast to the mincuts in [10])of a given graph.

1) MCCS Algorithm: We begin by setting theand its congestion to zero. A vertex of the graph that has the leastdegree of connectivity is selected as the seed vertex. We con-struct the BFS tree of the graph with the seed vertex as its root.If the seed vertex is not a cut vertex of the graph, the incidentedges to the seed vertex are a minimal cutset of the graph. Weuse Definition 9 to calculate the congestion of the minimal cutsetand compare the results with the congestion of MCCS. If theformer is greater than the latter, the minimal cutset is consideredas the new MCCS. The algorithm contracts the vertices from alist of the neighboring vertices of the seed vertex, the elementsof which have the BFS rank greater than the BFS order of theseed vertex. A contracted graph is then created by contractingthe seed vertex and one of the vertices in the list of neighboringvertices. The contracted vertex is the new seed vertex and itsBFS rank is set equal to the BFS rank of the last contractedvertex. The contraction and checking for cut-vertex is repeatedfor all contracted graphs recursively until all neighboring ver-tices of the seed vertex are scanned. The above algorithm scansall minimal cutsets of the network one by one and calculatesthe congestion of each cutset to find the most congested cutset,which is a tight lower bound of the chromatic number of the net-work. The computational complexity of this algorithm is linearper cutset [7], resulting in substantially reduced computationsfor large bounded degree mesh networks.

IV. COMPUTATIONAL RESULTS

To show that the MCCS algorithm obtains a tighter boundthan the LP-relaxation method, we used two sets of randomlygenerated connection request instances on NSFNet and EuroNet[12]. Each set consists of 20 problem instances. In each probleminstance of the first set (Set 1), the number of required full du-plex connections for each pair of nodes is either 1 or 2 withequal probabilities 0.5. In each problem instance of the secondclass (Class 2), the number of required full duplex connectionsfor each pair of nodes is drawn uniformly from the integers inthe range of 1–10. Lee et al. [12] obtained a lower bound for the

TABLE IAVERAGE LOWER BOUNDS FOR 20 RANDOMLY GENERATED

CONNECTION REQUEST INSTANCES

chromatic number in similar problem instances by the LP-relax-ation method. Table I compares the results. The average lowerbound of the MCCS method is greater than the resulting averagelower bound of LP-relaxation, which means that MCCS gives atighter lower bound.

V. CONCLUSION

We introduced the concept of cutset congestion for a graph,and showed that congestion of the most congested cutset isa tight lower bound for the load as well as for the chromaticnumber of the RWA problem. Using a novel and efficientmethod which we have developed in [7], we proposed theMCCS algorithm to find a tight lower bound for the chromaticnumber and the network load of any network and any trafficpattern. We also showed that MCCS results in a tighter lowerbounds than the LP-relaxation method.

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