DIVISION MULTIPLEXING OPTICAL NETWORKS
a thesis
submitted to the department of industrial engineering
and the institute of engineering and science
of b_
ilkent university
in partial fulfillmentof the requirements
for the degree of
master of science
By
Gunes Erdo~gan
opinionit isfullyadequate, inscop e and inquality,as a
dissertationfor the degreeof Master of Science.
Asst. Prof. Oya EkinKarasan (Sup ervisor)
I certify that I have read this thesis and that in my
opinionit isfullyadequate, inscop e and inquality,as a
dissertationfor the degreeof Master of Science.
Asst. Prof. Ezhan Karasan
I certify that I have read this thesis and that in my
opinionit isfullyadequate, inscop e and inquality,as a
dissertationfor the degreeof Master of Science.
Asso c. Prof. Mustafa Pnar
Approved for the Instituteof Engineeringand Science:
Prof. MehmetBaray,
NETWORK DESIGN PROBLEMS IN WAVELENGTH
DIVISION MULTIPLEXING OPTICAL NETWORKS
Gunes Erdo~gan
M. S. in Industrial Engineering
Supervisor: Asst. Prof. Oya Ekin Karasan
August 2001
In this study, we analyze the network design problem arising in Wavelength
Division Multiplexing (WDM) networks where trac is static, wavelength
interchanging is allowed and the lo cation and numb er of the wavelength
interchangers are to b e determined. Givenatop ology and tracdata, wetry to
ndthe b erandwavelengthinterchangercon guration withthe minimumcost,
thatcan establishallgivenconnections. Wepresentdi erentformulationsof the
problem and somevalid inequalities. Finally, we prop ose a heuristic metho d of
generatingfeasiblesolutions, applythemetho don threedi erenttop ologies with
varying tracdata, and presentthe results. Themetho d is basedon the idea of
partitioning the problem into two; routing problem and wavelength assignment
and interchanger lo cation problem. Our results prove to b e close to the lower
b ounds we generate, and indicate that the b er cost p erformance of the case
where all no des are wavelengthinterchangers can b e attained using a relatively
Ozet DALGABOYU B OL US UML U CO GULLAMA KULLANILAN OPT _ IK _ ILET _ IS _ IM A GLARINDA A G TASARLAMA PROBLEMLER _ I
Gunes Erdo~gan
Endustri MuhenlisligiYuksek Lisans
Tez Yoneticisi: Yrd. Doc. Oya Ekin Karasan
Agustos 2001
Bu calsmada, Dalga Bol us uml u Cogullama kullanlan aglarda, tra gin
duraganoldugu,dalgab oyudon us um uneizinverildigivedalgab oyudegistiricilerin
saysnnveyerlerininb elirlenmesininsozkonusu oldugu agtasarmproblemlerini
inceledik. Ag yaps ve tra k bilgisi verildigi halde, en az maliyete sahip olan
ve verilen baglantlar saglayabilecek bir ag tasarlamaya calstk. Problemi
ifade eden degisik form ulasyonlar ve baz gecerli esitsizlikler sunduk. Sonuc
olarak, olurlu coz umler uretmek icin bulgusal bir yontem onerdik, yontemi
farkl tra k bilgileri ile uc farkl ag yapsnda uyguladk, ve sonuclar sunduk.
Yontem problemi iki probleme ayrmak kri uzerine kuruludur: yol atama
problemi ve dalgab oyu atama ve dalgab oyu degistirici yeri saptama problemi.
Sonuclarmz, uretti gimiz alt snrlara yakndr, ve gostermektedir ki b ut un
d ug umlerindalgab oyudegistiricioldugudurumdaki b ermaliyetip erformansna,
Ag Tasarm,Cok
Iwouldliketoexpress mydeep estgratitudetoAsst. Prof. OyaEkinKarasan
for all the encouragement and trust during my graduate study. She has b een
sup ervising mewith patienceand everlastinginterest.
I am grateful to Asst. Prof. Ezhan Karasan for his invaluable guidance,
remarksand recommendations.
I amalso indebted to Asso c. Prof. Mustafa Pnar for accepting to read and
reviewthis thesis and for his suggestions.
Iwouldliketoexpressmydeep estthankstoOnurBoyabatlforhiscontinuous
moralesupp ort, friendship and for teaching meto have faithin myself.
Iwouldliketoextendmysincerethanksto,CumhurAlp erGelogullar,Seng ul
Dogan, FilizG urtuna, Cagr G urb uz and Cerag Pince,for their keen friendship
and helps.
I would also like to thank my brother
Ozg ur Kutluozen for his everlasting
supp ort, for the insights he gave me ab out life and for managing to make me
smileeverytime.
I would also like to thank Meral Kutluozen and Ertan Kutluozen for their
patience,love,and bringing meup fromasuburban b oy to a fullgrown man.
Finally, I would like to express my gratitude to Ebru Donmez for her love,
understanding and kindness. I owe so muchto her for makingmediscoverallof
Abstract i Ozet iii Acknowledgement vi Contents vii List of Figures ix List of Tables x 1 Introduction 1
2 Formulating the problem 9
2.1 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9
2.2 Minimalbinary formulation : : : : : : : : : : : : : : : : : : : : : 11
2.3 Stronger binary formulation : : : : : : : : : : : : : : : : : : : : : 12
2.4 Aggregated formulation: : : : : : : : : : : : : : : : : : : : : : : : 13
2.5 Valid inequalities : : : : : : : : : : : : : : : : : : : : : : : : : : : 16
2.6 Problems ab out the formulations : : : : : : : : : : : : : : : : : : 19
2.7 Pro of ofNP-Hardness : : : : : : : : : : : : : : : : : : : : : : : : 20
3 Exploring the subproblems 22
3.3 Generating strong lowerb ounds : : : : : : : : : : : : : : : : : : : 32
4 A solution method 35
4.1 Declaration of the overall pro cedure : : : : : : : : : : : : : : : : : 35
4.2 Remarksab out the pro cedure : : : : : : : : : : : : : : : : : : : : 36
4.3 Analysis of Results : : : : : : : : : : : : : : : : : : : : : : : : : : 37
4.4 NFSNET : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 39
4.5 ARPA2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 45
4.6 MESH32 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 51
3.1 Structure of constraintmatrixof IP4 : : : : : : : : : : : : : : : : 26
3.2 Structure of Dualof IP4 : : : : : : : : : : : : : : : : : : : : : : : 26
3.3 Structure of Mo di ed Dualof IP4 : : : : : : : : : : : : : : : : : : 27
3.4 Faces of asimplemeshnetwork : : : : : : : : : : : : : : : : : : : 33
3.5 Condensed graph of the graph in Figure3.4 : : : : : : : : : : : : 34
4.1 NFSNET top ology : : : : : : : : : : : : : : : : : : : : : : : : : : 39
4.2 ARPA2 top ology : : : : : : : : : : : : : : : : : : : : : : : : : : : 45
4.1 Lower Bounds for the NFSNET Top ology : : : : : : : : : : : : : 40
4.2 Results for the NFSNET top ology, KSP solvedto 6 alternatives : 41
4.3 PercentDeviations of KSP metho dfrom the lowerb ounds forthe
NFSNET top ology : : : : : : : : : : : : : : : : : : : : : : : : : : 42
4.4 Percent Deviations of Wavelength Assignment from the lower
b ounds forthe NFSNET top ology : : : : : : : : : : : : : : : : : : 43
4.5 WIXCRequirementsforthe NFSNET top ology : : : : : : : : : : 44
4.6 No de Frequenciesfor the NFSNET Top ology : : : : : : : : : : : : 45
4.7 Lower Bounds for the ARPA2top ology : : : : : : : : : : : : : : : 46
4.8 Results for the ARPA2 Top ology, KSP solved to 6 alternatives : : 47
4.9 PercentDeviations ofKSP Metho d fromthelowerb ounds forthe
ARPA2 top ology : : : : : : : : : : : : : : : : : : : : : : : : : : : 48
4.10 Percent Deviations of Wavelength Assignment from the lower
b ounds forthe ARPA2 top ology : : : : : : : : : : : : : : : : : : : 49
4.11 WIXCRequirementsforthe ARPA2 Top ology : : : : : : : : : : : 50
4.12 No de Frequenciesfor the ARPA2Top ology : : : : : : : : : : : : : 51
4.13 Lower Bounds for the MESH32 Top ology : : : : : : : : : : : : : : 51
4.14 Results for the MESH32 Top ology, KSP solvedto 6 alternatives : 52
4.15 PercentDeviations for the MESH32 Top ology : : : : : : : : : : : 53
4.16 WIXCRequirementsforthe MESH32 Top ology : : : : : : : : : : 53
Introduction
Computer networking has b een an imp ortant area of research for a long
time. With the tremendous growth of the Internet, sp eed and capacity
requirements for computer networks have increased considerably. Existing
network technologies did not seem to satisfy this huge requirement. This
was when all-optical networks came into the picture. All-optical networks
o ered higher sp eed, b etter reliability and more capacity than conventional
networks. All-optical networks are networks where information is converted
to light, transmitted as light, and reaches its nal destination directly without
b eing converted to electronic form in b etween. This metho d of transmission
of messages is sup erior to the previous metho ds. All-optical networks promise
data transmission rates several orders of magnitudes higher than the current
networks. The key to high sp eeds in these networks is to maintain the signal in
opticalformso as togetridofthe conversiontimefromopticalformto electronic
form and vice versa. All-optical networks are considered as the transp ort
networks of the future. The major applications for such networks are in video
conferencing,scienti cvisualization,real-timemedicalimaging,high-sp eedsup
er-computinganddistributedcomputing[6],[13],[15]. Tosolvethecapacityproblem,
WavelengthDivisionMultiplexingwasdevelop ed. Themostp opularapproachto
utilizethehigh-capacity ofall-optical networksisto divideopticalsp ectruminto
many di erent channels, each channel corresp onding to a di erent wavelength.
datastreamsto b etransferredconcurrentlyalongthesame b er-opticcable,with
di erent streams assigned separate wavelengths [13]. Although WDM increases
the capacity of all-optical networks, it also increases the complexityof network
management. Once a message is assigned a wavelength at its source no de,
this assignment cannot b e changed at subsequent no des. Networks which only
encountered capacity blocking until now, are subject to a new typ e of blo cking
called wavelength blocking. In the former, a message cannot b e delivered to its
destinationb ecause allpathstodestination areblo ckedby linksthat areused by
other messages. In the latter, a message cannot b e delivered to its destination
b ecauseevenifthereexistsapathtothedestination,nowavelengththatisunused
on all links along the path can b e found. To overcome this problem, devices
that can change the wavelength assignment of a connection are used. These
devices are referred as wavelength interchangers in this study. Much research
has b een done to investigate the e ectsof wavelength interchangers on routing,
numb er of wavelengths required, blo cking probability, throughput etc. Most of
the approaches consideredeitherno wavelengthinterchanging,calledWavelength
Path Scheme(WP); or wavelengthinterchangingcapabilityat each no de, called
Virtual WavelengthPathScheme(VWP).Theproblemofdeterminingtheroute
and wavelength assignment of each connection in a WDM network is known as
the Routing and Wavelength Assignment(RWA) problem. RWA problems have
twomaincategories,staticanddynamic. Intheformer,allconnectionsareknown
a priori, whereas in the latter connection requests arriverandomly.
WDM networks receivedconsiderable interest from researchers. Raghavan
andUpfal(1994)studiedroutingasetofrequests(eachofwhichisapairofno des
to b econnected bya path)using a limitednumb erof wavelengthsensuring that
di erentpathsusingthesamewavelengthneverusethesamephysicallink. They
presented routing techniquesand established connectionsb etweentheexpansion
of anetwork and the numb erof wavelengthsrequired for routing on it [6].
Ramaswami and Sivarajan (1995) studied maximizing the amount of
dynamic trac carried when there is a single b er on each link and wavelength
conversion is not allowed. They presented an IP formulationand proved upp er
b ounds for b oth the IP and the LP that corresp onds to its relaxation. They
that if allno des havethe capability ofinterchangingwavelengthassignment,use
of total capacity of the wavelength division multiplexing can b e improved by
10-40% [9].
Wauters and Demeester (1996) presented formulations for maximizing the
carried trac for two cases; when wavelength interchanging is not allowed,
and when every no de is a wavelength interchanger. They used two kinds of
formulations, namely ow and path formulations. While their ow formulations
werequitecloseto theusualnetworkformulations,pathformulationswerebased
on enumeratingp ossiblepathsb etweensourceanddestinationpairsandcho osing
oneamongthem. Finally,theypresentedaniterativeheuristicRWAalgorithmto
minimizethenumb erofwavelengthsrequiredtosuccessfullydelivereachmessage
to its destination. The algorithm was based on p erforming lo cal search on an
initial routing and wavelengthassignment. At each iteration,the path with the
largest wavelengthnumb er(or all the paths that interfered with it) was tried to
b e rerouted on a smaller wavelength numb er. Results of their exp erimentation
suggested that wavelengthconversiondidnot makeasigni cantreduction inthe
numb erofwavelengthsrequired,andwavelengthinterchangingwasnotnecessary
at every no de. They also concluded that wavelengthinterchanging capability at
somesp eci c no des may b e enough to overcomewavelengthblo cking [10].
Nagatsu, Okamotoand Sato (1996) prop osed algorithmsfor RWA problem
in a multi- b erenvironment(more than one b er can exist b etweentwo no des)
for b oth WP and WVP schemes. Their algorithm for the VWP scheme was
aimed at minimizingthe b er requirement,whereas their algorithm for the WP
schemewas aimedat minimizingthe numb erofwavelengthsrequired. Theyalso
prop osed algorithms for failure restoration in VWP and WP schemes, in which
they considered single-link-failures. Theyconcluded that the di erence b etween
VWP and WP schemesincreasedas the numb erof wavelengthsincreased[11].
BanerjeeandMukherjee(1996) studiedRWAfor staticanddynamictrac
in single- b er WDM networks. They partitioned the RWA problem into two
stages, rst routing and then wavelength assignment. First problem was the
wellknownmulticommo dity owproblem. Theymanagedto obtainresultsclose
to the LP lower b ound for the multicommo dity ow problem, using a heuristic
theproblemintothegraphcoloringproblemusingtherouting theyobtained,and
used smallest-last algorithm to minimizethe numb er of wavelengths used. The
algorithm basically starts by coloring the no des with the maximumdegree, and
continues withcoloring the smallerdegree no des. Theirresults werecloseto the
LP lowerb ounds [12].
Bermond et al. (1996) presented upp er and lower b ounds for the numb er
of wavelengthsrequiredto \gossip" (one-to-allcommunication)and \broadcast"
(all-to-all communication) when each link has only one b er, and wavelength
interchangingisnotp ossible,innetworkswitharbitrarytop ologiesandparticular
networksof interestsuchas ring, torus, hyp ercub e[13].
Armitage,Cro chatandLeBoudec(1996)presentedatabusearchalgorithm
for the WP scheme,namelyDisjointAlternatePath(DAP), that nds a routing
minimizingthe numb erof brokenconnections incase ofa single-linkfailure[14].
FlamminiandScheideler(1997)studiedroutingasetof\dynamic"requests
with a limited numb er of wavelengths, single b er on each link, and without
wavelengthconversion. They suggested a proto colfor routing, and appliedtheir
results to di erenttop ologies [15].
Qiao, Mei,Yo oand Zhang(1998) suggested slicingan optical networkinto
several Virtual Optical Networks (VONs) and equipping each VON according
to its trac structure. They concluded that VONs supp orting dynamic trac
require asmallnumb erof wavelengthsand use of wavelengthinterchangers, but
VONs supp orting static trac require a larger numb er of wavelengths and no
wavelengthinterchangers[16].
Ramamurthy and Mukherjee (1998) presented a review/survey of the
underlying technologies ofWDM, WDMnetworkdesign metho dsand analytical
mo dels used in wavelength-interchangeablenetworks. One of the questions they
p osed was: \Aninterestingquestion whichhas not b eenanswered thoroughly is
where (optimally) to place these few converters..." [20]. One of the outcomesof
this thesis workis an algorithmto answerthis question.
ZhangandQiao(1998)studiedwavelengthassignmentfor\dynamic"trac
in multi- b er WDM networks and presented an algorithm, namely Relative
Capacity Loss algorithm,to minimizethe probability of blo cking. They claimed
algorithm[21].
Alanyali and Ayanoglu (1998) presented two heuristics for routing and
wavelengthassignmentofasetofstaticconnectionrequestsinWPscheme. First
heuristic was aimed at minimizing the total weighted b er length and did not
consider fault tolerance, while second heuristic was an adaptation of the rst
heuristic for the faulttolerantcase and considered several failurescenarios [22].
Yuan et al. (1998) assessed b ene ts of wavelengthconversion and claimed
that wavelength conversion could result in an increase of throughput in a
environmentunderdistributed control [23].
Qiaoand Mei(1999) studiedtheminimumnumb erofwavelengthsrequired
p er link for a givennetworkto b e rearrangeably non-blo cking in WP and VWP
schemes. They claimed that WP and VWP p erformed equivalently in linear
array top ologies, whileVWP p erformedslightly b etterinrings, meshes,tori and
hyp ercub es[25].
Yates, Rumsewich and Lacey (1999) presented a review of p erformance
improvements o ered by wavelength interchanging. They also discuss the
e ects of the top ology, numb er of wavelengths, and RWA algorithms on the
p erformance improvements of wavelength interchanging. They concluded that
in mostnetworks,wavelengthinterchangingcapabilitydo es result inamo derate
improvement in p erformance. On the other hand, when path lengths are large
and interference lengths are small, wavelength interchangers can result in a
considerable increase of p erformance. They also concluded that wavelength
interchanger capabilityat alimitednumb erof no desusually p erformsequivalent
to the case where every no de is awavelengthinterchanger [26].
Subramaniam,Azizoglu and Somani(1999) studied the problemof nding
the optimal placement of a given numb er of wavelength interchangers in the
network, when the o ered trac is dynamic. They presented a dynamic
programming algorithm to nd the optimal placement of interchangers on a
path, when link loads are nonuniform. Their results showed the imp ortance of
wavelength conversion. Optimallyplaced 4 interchangers on a 11-no de, 10-edge
path resulted in a reduction of blo cking probability by more than two orders of
magnitude [27].
formulations corresp onding to three di erent objective functions. First was to
maximize sum of (utilization of no de i * numb er of wavelength interchangers
at no de i) over all no des. Second was to maximize the pro duct of (utilization
of no de i * numb er of wavelength interchangers at no de i) over all no des. The
last objective was to maximize the minimumof (utilization of no de i * numb er
of wavelength interchangers at no de i) over all no des. They used dynamic
programming to solve rst two problems, and a greedy algorithm to solve the
third problem whichwas provento nd the optimal[28].
Park, Shin and Lee (1999) prop osed algorithms for routing and minimum
wavelength requirement when routing is known. For wavelength interchanger
lo cation they simply recommended to allo cate them to no des in descending
order of numb er of paths passing through numb er of no des, until feasibility is
attained. Finally,theygaveanoptical b erdimensioningalgorithmtodetermine
the numb erof b ers on eachedge required for feasibilityof ow [29].
Xiao,LeungandHung(2001)prop osed analgorithm,namelytheTwo-stage
Cut Saturation Algorithm, for designing an all-optical network with minimum
cost. Theyconcludedthattheiralgorithmp erformedfairlywellandifwavelength
interchanging is allowed on all no des, total cost of links may b e reduced ab out
20% [30].
Inthe literature,WDMnetworkdesignproblemhas manydi erentmetrics
suchas throughput, blo cking probability, numb erof wavelengthsrequired, total
b er length used, reliability, control complexity, etc. To the b est of our
knowledge, minimum numb er of wavelength interchangers and their optimal
lo cation is a problem that is virtually untouched. A few studies fo cus on
optimally placing a limited numb er of wavelength interchangers on a network,
in order to minimize blo cking probability, or minimize the total b er cost,
but with given routing data. Actually, in a hybrid network comp osed of
wavelength interchanging and non-interchanging no des, RWA problem b ecomes
harder, b ecause wavelength assignment of a transmission may or may not b e
changed according to its routeand the lo cationof the wavelengthinterchangers.
The actual overall problem is to design a minimum cost network while solving
the corresp onding RWA problem simultaneously, given the trac data and the
network design problem with minimumtotal b er and wavelength interchanger
costs, iscarried out. Before statingthe problem,factsab outthe structure of the
communication networkweanalyze willb e presented. Weare givena connected
graph with n no des and m edges. Each no de transmits and/or receives data.
Eachconnectionisassigned awavelengthat itssource no de. Eachno de iseither
a Wavelength Interchanger Cross-Connect (WIXC) or a Wavelength Selective
Cross-Connect (WSXC), where the former has the capability of changing the
wavelengthassigned to a connectionexpressingthrough theno de, and the latter
do es not have this capability. The former has an undetermined cost, since it is
not commerciallyavailableat the timethis thesis is submitted. The latter has a
cost, but it is out of consideration, b ecause each no de requires one to transmit
and receivemessages. Atleastone b ershouldb einstalledon anedgeiftheedge
willb e used. Fib ers are unidirectionaland each b ercan accommo date onlyone
message of each distinct wavelength. So if two messages are assigned the same
wavelengthand owb etweenthe sametwono des,then at leasttwo b ersshould
b e installed on that edge. We assume that each link has a variable cost p er
b er installed but no xed cost of installation. Throughout all formulations in
this thesis, it is assumed that capacity just enough to accommo date the ows
is necessary and sucient. Providing extra capacity for reliability is out of
consideration.
The problem can b e stated as follows: Given a particular top ology and
trac data, determine the con guration of b ers to b e installed, numb er of
wavelength interchanger devices and their lo cations, routing and wavelength
assignmentofeachconnectionateachlinkituses,suchthattheresultingnetwork
has the minimumcost.
In Chapter 2, various IP formulations of the problem are presented. First
formulationis a binary formulationwhere connections are represented as binary
variables. Secondformulationisanotherbinaryformulationwhere\interchanger"
constraints are stronger. Final formulationin Chapter 2 is an aggregated mo del
where connectionsareconsolidatedaccording totheir sourceno des. Aggregation
greatly reduces the numb er of variables, but some valid cuts which exploit the
binary structure of the problem cannot b e added. Next, valid cuts prop osed for
are, 1) the integermulticommo dity owproblemwith variable upp er b ounds, 2)
wavelengthassignmentproblem,and 3)interchangerlo cationproblem. Hardness
ofthesubproblemsarediscussedandIPformulationsforeachofthesubproblems
are presented. Aworstcasecostb ehaviourexpressionisderivedforshortestpath
routing. A metho d of generating strong lower b ounds for our problem using the
multicommo dity ow problemwith variable upp er b ounds is also presented.
In Chapter 4, a solution metho d is prop osed. The metho d can
b e summarized as follows: First, the problem is relaxed into an integer
multicommo dity ow problem with variable upp er b ounds. Second, a feasible
solution for the case without any wavelength interchangers is generated using
the solution from the rst stage. More sp eci cally, in stage two, routing found
duringthe rststageis xedandthewavelengthassignment(withoutwavelength
interchangers) problem is solved with the sp eci ed routing. If the cost of
second stage is greater than the cost of the rst stage, a third problem of
wavelengthinterchanger placementis solvedto determinehow manywavelength
interchangers are required and where they should b e placed. Next, the results
of the prop osed metho d are presented and analyzed. The results are obtained
by applying the metho d to three di erent top ologies with randomly generated
trac data and varying levelsof trac density. The results are compared with
the lower b ounds generated using the metho d describ ed inChapter 3. The fact
that the b er cost p erformance when each no de is a wavelength interchanger
can b eattained by arelativelysmallnumb erof wavelengthinterchangers, is our
most imp ortant contribution to the literature. The lo cation of these wavelength
interchangersdep end b oth on the top ology andthe trac,but most ofthe time,
they happ en to b elo cated in the `middle'of the graphor at the `crossroads'.
The lastchapteristhesummaryofthe thesis. Resultsare summarizedand
Formulating the problem
To have a b etter understanding of the structure of the problem, a precise
mathematical expressionof the problemis required. Although theyare not very
useful insolving the problem itself, the formulations of the problem o er many
insightsab outwaysofsolvingtheproblem. ThreeIPformulationsoftheproblem
withdi erentstrongandweakp ointsarepresentedinthischapter. Beforemoving
on to the formulations,it isnecessary to state the notation to b eused.
2.1 Notation
Let G=(N;E) b e the graph corresp ondingto the networktop ology where
N isthesetofno des, N =f1;:::;jNjg,andE isthesetofedges, E =f1;:::;jEjg.
Let K b e the set of connection demands, with cardinality jKj. Each element
of the set is a tuple (s
k ;d
k
), where s
k
denotes the source and d
k
denotes the
destination of demand k.
Let W b e the setof wavelengthsavailable,W =f1;2;:::;jWjg.
Let x
ijk w
b e the binary variable representing the owof connection k fromno de
x ijk w = > > > < > > > :
1 if demandk ows fromno de ito no de j with
the wavelengthassignment w
0 otherwise
Let f
ij
b e the numb erof b ers to b e installed b etweenno des i and j.
Let a
i
b e the binary variable representing the existence of a WIXC. In other
words, a i = 8 < : 1 if there is aWIXCat no de i 0 if there is aWSXCat no de i
Let HC b e the di erence b etweenthe cost of a WIXCand the cost of a WSXC.
Let c
ij
b e the cost ofinstalling one b erb etweenno des i and j.
Let y
ik
b e the parameter for the demand/supply of the network ow. In other
words, y ik = 8 > > > < > > > :
1 if no de i is the source no de of demand k
1 if no de i is the destination no de of demand k
Based on the de nitions ab ove,the formulationis as follows: (IP1) Min P (i;j)2E c ij f ij +HC( P i2N a i ) s.t. P j2N P w 2W x ijk w P j2N P w 2W x jik w = y ik 8i2N;k 2K NC P k 2K (x ijk w +x jik w ) f ij 8(i;j)2E;w2W BC P j2N P w 2W w(x jik w x ijk w ) (jWj 1)*a i 8i2N;k 2K where y ik =0 IC1 P j2N P w 2W w(x ijk w x jik w ) (jWj 1)*a i 8i2N;k 2K where y ik =0 IC1 x ijk w 2B f ij 2I a i 2B
Numb erof variables: (2jEjjKjjWj)+jEj+jNj
Numb erof constraints: (3jNjjKj)+(jEjjWj)
InIP1,NetworkConstraints(NC)providetheconservationof ow. Bundle
constraints (BC) make sure that enough numb er of b ers are deployed on an
edge to accommo date the demand through that edge. Interchanger constraints
(IC1&IC2)ensurethatif amessagechanges itswavelengthat no dei,thenno de
i must b e a WIXC. Supp ose a connection arrives at a no de i with wavelength
assignmentw 1
andcontinuestoneighb ouringno dej withwavelengthassignment
w 2
. This resultsinadi erenceb etweenthewavelengthassignmentof in owand
out ow, and IC1 and IC2 forces a
i to b e at least jw 1 w 2 j jWj 1 . Since a i values are
constrained to b e binary, this means that a
i
= 1. Strong p oint of formulation
IP1 is that it states the problem with minimumnumb er of constraints that the
author could. Another advantage is that IC1 and IC2 are comp osed of binary
variableswith non-binary co ecients. This prop ertymayb e used forgenerating
IP1 can b e further strengthened with stronger interchanger constraints.
Assume that a message comes to no de i with wavelength assignment w and
leaveswith wavelength assignment w+1. In this case, interchanger constraints
of IP1 will force the interchanger assignment variable a
i
to b e at least 1
jWj 1 .
The stronger interchanger constraints prop osed are:
P j2N x jik w P j2N x ijk w a i 8i2N;k2K ;w2W;w her ey ik =0 IC'1 P j2N x ijk w P j2N x jik w a i 8i2N;k2K ;w2W;w her ey ik =0 IC'2
If this new set of interchanger constraints are used, the interchanger
assignment variable a
i
will b e forced to b e 1, which shows that this set of
interchanger constraints are stronger. The price of strength is the increased
numb er of constraints. Also, ecient cover cuts cannot b e generated for this
formulation b ecause the co ecient matrix is comp osed of 0's and 1's. Second
formulationis as follows: (IP2) Min P (i;j)2E c ij f ij +HC( P i2N a i ) s.t. P j2N P w 2W x ijk w P j2N P w 2W x jik w = y ik 8i2N;k 2K NC P k 2K (x ijk w +x jik w ) f ij 8(i;j)2E;w2W BC P j2N x jik w P j2N x ijk w a i 8w2W;i2N;k2K where y ik =0 IC'1 P j2N x ijk w P j2N x jik w a i 8w2W;i2N;k2K where y ik =0 IC'2 x ijk w 2B f ij 2I a i 2B
Numb erof constraints: (jNjjKj)+(jEjjWj)+(2jKj(jNj 2)jWj)
2.4 Aggregated formulation
Two formulations presented have strong relaxations, which can b e further
strengthened with valid cuts. Unfortunately, for a small sized problem of 14
no des, 21 edgesand 60 connections, with8wavelengthsavailable,the numb erof
variablesrequiredis 20195 forb oth formulationspresented. For largerproblems,
the numb erofvariables b ecomesto o large forcommerciallyavailableMIPsolver
software. To overcome this problem, we used aggregation [18], [24], which is
simplyconsolidatingtheconnectionsaccordingtotheirsourceno des. Aggregated
connection requests are referred as commo ditiesfor the rest of this study. Since
consolidation is made using no des, clearly, numb er of commo dities b ecomes at
mostjNj. Forthemaximumreductioninthenumb erofvariables,weuseasimple
minimal cover formulation that ensures that the source or destination no de of
each connection is selected to b e a commo dity, and minimizes the numb er of
commo dities. Note that establishing a connection from no de i to no de j is no
di erent than establishing a connection from no de j to no de i, which in turn
means that source and destination no des of all messages can b e rearranged so
that the source no de of each connection is a commo dity, without changing the
problem. Once aggregation is done, connection set K b ecomes commo dity set
K' (which is a subset of the no de set N), and ow parameter y
ik
b ecomes the
aggregated owparameterY
ik
. Also,binary owvariablesb ecomegeneralinteger
variables. The metho d is quite useful for reducing the numb erof variables, for
example, numb er of variables required for the example ab ove b ecomes at most
4739. But since we discard the binary structure ofthe problem,somevalid cuts
exploitingthebinarystructureofthe originalproblemareno longeruseful. Valid
cuts will b e discussed later in this chapter. Before presenting the aggregated
Givenno de setN and connection setK.
1. Solve minimum set cover problem to select a set K 0 = fc 1 ;c 2 ;:::;c jK 0 j g
of no des with minimum cardinality suct that either the source or the
destination of each connectionin K ispresentin K'.
2. For k:=1 to jKj if s k = 2K 0 then interchanges k and d k 3. For k:=1 to jK 0 j For i:=1to jNj if i=c k , Y ik
:=(numb erof elementsof setK with source c
k )
else,Y
ik
:= -(numb erof elementsof setK with source c
k and
destination i)
The aggregated formulationis as follows:
(IP3) Min P (i;j)2E c ij f ij +HC( P i2N a i ) s.t. P j2N P w 2W x ijk w P j2N P w 2W x jik w = Y ik 8i2N;k2K 0 NC P k 2K (x ijk w +x jik w ) f ij 8(i;j)2E;w2W BC P j2N x ijk w P j2N x jik w M*a i 8w2W;i2N;k2K 0 where Y ik 0 IC x ijk w 2I f ij 2I a i 2B
Numb erof variables: (2jEjjK 0 jjWj)+jEj+jNj Numb erof constraints: (jNjjK 0 j)+(jEjjWj)+(jK 0 jjN 1jjWj)
mo di ed to cop ewith the aggregated ow structure. What theystate is simply:
If, at a no de other than the source no de of the commo dity, the out ow of the
commo dityforawavelengthassignmentismorethanthein owofthecommo dity
with the particular wavelength assignment, then the no de is a wavelength
interchanger. Clearly, this means that the ow was assigned a wavelength it
was not assigned b efore. This time, we cannot state the second part, since we
haveaggregated the ows accordingto theirsourceno des,theirdestinationsmay
b e di erent. Hence, the out ow of a commo dity with a wavelengthassignment
may b e less than the in ow of the commo dity with the particular wavelength
assignment at a no de other than the source no de of the commo dity. Structure
of the networkconstraintsandthe bundleconstraintsare the sameas the binary
formulations presented b efore.
Aggregationhasb eenused intheliteratureforavarietyofmulticommo dity
ow problems. Due to the fact that the p erformance of branch & b ound
relies on the sp eed of the simplex algorithm, smaller numb er of variables grant
a considerable advantage to the aggregated formulations. Gendron, Crainic
and Frangioni (1998) p oint out that ([19]), LP relaxations of the aggregated
formulations for multicommo dity ow problems are much easier to solve, but
it is also more dicult to identify inequalities that tighten the lower b ound.
During the study, aggregation was used on multicommo dity ow formulations
for generating lower b ounds due to the improvement of sp eed it o ers. Even
with aggregation, numb er of variables b ecameto o large to handle for a 32-no de
50-link top ology due to the large numb erof no des. Although this formulationis
not as degenerate as the previoustwomo dels,LPrelaxationisto o weakand the
problem app ears to have a symmetric structure which reduces the eciency of
the branch &b ound. Thus,valid inequalitiesto tighten the lowerb ound and to
In this section, valid inequalities from the literature and valid inequalities
prop osed by the author are presented. Before stating the rst set of valid
inequalities,somede nitionsarerequired. LetS andT b esubsetsof no desetN.
Furthermore, letT =N nS. Let D
ST
b e the amount of trac the network has
to carry b etween partitions S and T. Let E
ST
b e the set of edges that connect
partitions S and T. Then,
(VI1) P (i;j)2E S T f ij d D S T jWj e 8S;T N;T =N nS
This rst set of valid inequalities is known as the cutset inequalities in
the literature. T. Magnanti, P. Mirchandani and E. Vachani have shown that
the cutset inequalitiesare facet de ning for the two-facility capacitatednetwork
loading problem(TFLP) ,whensubgraphsde nedbySandTareconnectedand
D
ST
> 0 [7]. TFLP problemis the problem of designing a capacitated network
with zero ow costs, where facilities of xed capacity can b e installedon edges.
Two typ es of facilities with di erent capacities and costs are considered. The
problemisquitesimilartoarelaxationthatwillb eusedto generatelowerb ounds
in Chapter 5. In fact, the only di erence b etweenthe two problems is that the
formulationwe will present allows only one typ e of facility. Althoughwe do not
givea pro ofthat the cutset inequalitiesare facetde ning for our problems,they
proved to b e very strong during the exp erimentation. The problem ab out the
cutset inequalitiesis that, they are exp onentialin numb er. Every subset S of N
that satis es1jSjb N
2
cgivesaprobablecut(so thatTwillcoverthesubsets
with greater numb erof elements). Total numb erof probable cutset inequalities
is 2 jNj 1 1 if jNj is o dd and 2 jNj 1 + ( jNj jNj=2 ) 2 1 if jNj is even. Enumerating
all probable cutsets (also checking each and every one of the probable cutsets
for connectivity of S and T) is not feasible for networks with more than 20
no des. Note that each no de constitutes a connected S set. Likewise,every edge
subsets of N. Numb er of constraints that can b e generated in this manner is
jNj+jEj, and these cuts can b e used to strengthen allthree formulations.
Analysis of optimum solution of the LP relaxation of formulation IP2 by
barrier metho d of CPLEX, motivated the author to nd the second set of valid
cuts thatwillb e presented shortly. Theoutput suggested that barrieralgorithm
divided the ow uniformly b etween all wavelengths and sent the divided ows
through the shortest paths. Becauseof the structure of the bundle constraints,
this kind of ow could only increase the b er requirement by 1
jWj
. What the
problemrequiresissimply: Atleastone b erisrequiredifoneunitof owpasses
through an edge. To state this in terms of the formulation,the following set of
valid inequalitieshave b eenintro ducedto the mo del.
(VI2) P w 2W (x ijk w +x ijk w ) f ij 8(i;j)2E;k 2K
Numb erof valid inequalities: (jEjjKj)
Actually, this set of valid inequalities can b e extended to cover a larger
numb er of connections. The statement ab ove can b e restated as : At least
n + 1 b ers are required if n jWj + 1 units of ow pass through an
edge. Unfortunately, for a jKj connection problem where jWj wavelengths
are available, the corresp onding numb er of valid inequalities generated for
values of n larger than 0 are: jKj jWj+1 jWj for n = 1, jKj (2jWj)+1 jWj for
n = 2, and so on. Adding this many constraints expands the problem to o
muchb eyond tractability. However, valid inequalities generated by considering
single connection case provided considerable tightening of lower b ound during
exp erimentation. Another advantage is that they are p olynomial in numb er.
Unfortunately, they decrease the sp eed of the simplex algorithm dramatically.
Moreover, they dep end on the binary ow structure, thus, they cannot b e used
structure of an IP may cause branch-and-b ound to p erform p o orly b ecause the
problem barely changes after branching [17]. All of the formulations presented
up tonowhaveasymmetricstructure. Infactgivenanoptimalsolution,jWj! 1
optimal solutions can b e generated. A simplepro of of the previousstatementis
as follows:
Proposition 1 Given an optimal solution, jWj! 1 optimal solutions can be
generated.
Proof: Givenan optimal solution x
, each ow has a wavelengthassignment
for each edge it ows on. Notice that a wavelength assignment is only a label.
Changingthenameofalab elwillnotresultinanychangeintheoptimalsolution
value. Givenan optimallab elinganditscorresp ondingpartitioningof ows(jWj
sets of ows), one can interchange the names of the lab els without disrupting
the optimality. So given jWj lab elsand jWj partitions, total numb erof p ossible
one-to-onematchingsisjWj!. Sincewearegivenoneofthe assignments,numb er
of distinctassignmentsthat can b e generatedis jWj! 1 . 2
To decrease this level of symmetry, the following set of valid inequalities is
prop osed. (VI3) P (i;j)2E P k 2K x ijk w 1 P (i;j)2E P k 2K x ijk w 2 8w 1 ;w 2 2W;w 1 =w 2 +1
Numb erof valid inequalities: jWj 1
VI3 simplystates that the most crowded wavelengthshould b e jWj, next
most crowded wavelength should b e jWj 1, and so on. As opp osed to the
valid inequalities presented up to this p oint, this third set of valid inequalities
do not tighten the lower b ound, instead, they shrink the branch-and-b ound
tree. Although very small in numb er,these inequalities shrinkthe search space
considerably, but duringthe exp erimentations,itwas noticed thatthey decrease
of the simplexalgorithmto o muchto b e useful.
2.6 Problems about the formulations
The rst major problem ab out the formulations is the huge numb er of
variables. For a large problem of 32 no des, 50 edges and 100 connections,
with 8 wavelengths available, rst two formulations require 80082 variables
(80032 binaryvariablesand50 integervariables),whereas thethird (aggregated)
formulation requires12882 variables on the average (12832 binary variables and
50 integer variables), assuming that on the average half of the no des will b e
selected as commo dities. This many variables result in longersolution timesfor
the LP relaxation, longer dual simplex times at each no de of the
branch-and-b ound treeand larger branch-and-b ound trees, and consequently huge timeand
memoryrequirements.
Second problem ab out the formulations is the high degree of degeneracy,
esp ecially for the rst two formulations. It is known that LP relaxations of
capacitated network design problems are often highly degenerate [19]. It is
observed that the objective value of the relaxation tends to stall for a few
thousands ofiterations at atime. Afew hundred,andfrequentlyafewthousand
dual simplex iterations are required for re-optimization at each no de of the
branch-and-b ound tree. The exp erimentations has b een made using ARPA2,
NFSNET and a 32 no de top ology representing long distance telephone network
with no des corresp onding to major US cities. For the rest of this study, this
32 no de top ology will b e referred as MESH32. Even for the smallest network
structure (NFSNET) used for exp erimentation, no optimal result could b e
obtained for a reasonable numb er of connections using the three formulations
ab ove.
Third problem ab out the formulations is a general problem ab out the
capacitated network design problems. As Magnanti, Mirchandani, and Vachani
p oint out: \In general, linear programming lower b ounds are weak for most
capacitated networkdesign problems..." [7], [19]. Eventhough branch& b ound
(VI2) to formulations (IP1) and (IP2) resulted with an increase of ab out 5% in
the lower b ound, but after a week of computer time, the author had to give up
b ecause ofthe unsatis ablememoryrequirementof the branch-and-b ound tree.
2.7 Proof of NP-Hardness
S. Even, A. Itai and A. Shamir proved that the integral multicommo dity
ow problem is NP-Complete [1]. A simple transformation from integral
multicommo dity ow problem will b e presented to prove that our problem is
NP-Hard.
Theorem 1 The problem of routing, wavelength assignment and wavelength
interchanger location with minimum total cost is NP-Hard
Proof: Weknowthatthe integralmulticommo dity owproblemisNP-Hard [1].
The following formulation describ es the integral multicommo dity ow problem.
Note that u
ij
denotesthe upp er b ound on total ow on edge (i;j).
Min P k 2K P (i;j)2E c ijk x ijk s:t: P j x ijk P j x jik = y ik 8i2N;k2K NC P i;j (x ijk w +x jik w ) u ij 8(i;j)2E;w2W BC x ijk 2B
If the upp er b ounds u
ij
are replaced by variables (f
ij
) with corresp onding costs
(c
ij
), the resulting problem is also NP-Hard since the integral multicommo dity
ow problem is a sp ecial case of the resulting problem where f
ij = u ij . Now assume wereplacef ij with jWjf ij
. We know that the problemis NP-Hard for
jWj = 1, and the problem will b e no easier for other (p ositive) values of jWj.
Finally, notice that the sp ecial case of our problem where all no des are set to
b e wavelength interchangers, is equivalent to the integral multicommo dity ow
problem whereupp er b ounds are replacedwith jWjf
ij
. Consider the following
transformation: take the network for an integral multicommo dity ow problem
where upp er b ounds are replaced with jWj f
ij
capacity of a single link (jWj). Set all no des to b e wavelength interchangers.
Set the cost of an interchanger to 0. Set the b er costs to b e equal in b oth
problemsandthe owcostofaconnection foreachwavelengthassignmenton an
edge to b e equal to the owcost of that particular connection on that edge. In
this case an optimal solution to the former problemcan b e transformed into an
optimalsolutiontothelatterinthefollowingmanner. Foreachedge,constructan
arbitrarylistof ows onthat edge(inb oth directions). Assign(imo djWj+1)th
wavelength to the i'th ow on the list. This corresp onds to an optimal solution
to the latterproblem,sincewedo nothaveto carefor wavelengthcontinuityand
the ow and b er costs are equal. Similarly, taking an optimal solution of the
latter problemand placing each ow (not caring for the wavelengthassignment
of the ow) to itscorresp ondingedgeresultsinanoptimalsolution to theformer
problem. Clearly, the transformation is p olynomial. So, we can conclude that
Exploring the subproblems
Failure to optimize the whole problem, together with the pro of of
NP-Hardness, led the author to search for a \go o d" metho d of generating feasible
solutions. The metricof b eing go o d is, of course, the distance to a lower b ound
whichmightb egeneratedbyarelaxationoftheoriginalproblem. Topresentmore
than asimplegreedyalgorithm,ab etterunderstanding ofthemainproblemwas
required. Inordertounderstandthe grandproblem,subproblemswereidenti ed.
The subproblemsone can identifyare the routing problem,the wavelength
assignmentproblem andthe wavelengthinterchanger lo cationproblem. Routing
of connections may b e xed to solve wavelength assignment and wavelength
interchanger lo cation problems simultaneously. Wavelength interchanger
lo-cations (hence, numb er of wavelength interchangers) may b e xed to solve
the RWA problem of the resulting top ology. However, wavelength assignment
cannot b e xed, b ecause it dep ends on b oth the routing and the wavelength
converter lo cations. Thus, the author decided that the most appropriate way
to partition the problems is to nd a routing (without wavelength assignments)
that minimizes the b er cost, and then solve the wavelength assignment and
wavelength interchanger lo cation problems simultaneously. In the rest of this
chapter,these subproblems are analyzed.
3.1 Integral Multicommodity Flow Problem
To simplify the problem, wavelength assignment obligation was dropp ed
the problem was equivalent to the integer multicommo dity ow problem with
variableupp erb ounds,whichisaprovablyhardproblem. Notethatthisproblem
is a relaxation of our problem. The formulation which is equivalent to integral
multicommo dity ow problemwith variable upp er b ounds is as follows:
(IP4) Min P (i;j)2E c ij f ij s.t. P j2N x ijk P j2N x jik = y ik 8i2N;k 2K NC jWj*f ij P k 2K (x ijk +x jik ) 0 8(i;j)2E BC x ijk 2B f ij 2I
Numb erof variables: (2jEjjKj)+jEj
Numb erof constraints: (jNjjKj)+jEj
Di erent than the formulations presented b efore, subscript w for the ow
variables have b een eliminated. This formulation, to o, may b e aggregated
using the algorithm describ ed in Chapter 2. Detailed studies ab out solving
multicommo dity ow problems b oth heuristically and optimally have b een
done ([3],[5],[18],[24]). Using Lagrangian relaxation of the multicommo dity
ow problem is recommended (relaxing the bundle constraints so that the LP
relaxation ofthe restof the problem givesintegralresults)to pro ducea solution
close to the LP lower b ound ([4]). Both Lagrangian relaxation and Lagrangian
Dualizationhasb eenappliedtotheformulations,withoutsuccess. Unfortunately,
neither lower b ounds improved,norgo o d feasiblesolutions could b e generated.
Banerjee and Mukherjee tackled this problem using \Randomized
Round-ing", which uses the optimum solution of the LP relaxation to construct a
feasible solution ([12]). Instead of a probabilistic rounding metho d, a partial
column generation approach is prop osed by the author. Please note that a
one alternativefor eachsource-destination pair. Also note that total numb er of
feasible paths available increasesexp onentially with the numb erof no des in the
network. Author'sintuitionsuggested thatinan optimalsolution,shortest paths
were used much more frequently than longer paths. Hence, enumerating the
rst Ashortest paths forthe alternatives,andselectingamongthesealternatives
seemed to b e an ecient way to handle this problem. Many algorithms exist
for ndingthe k-shortest paths (KSP)ina networkwith nonnegative owcosts.
For a comparative study of existing studies on KSP, see [8]. Yen's algorithm
was implemented in C and used for nding the k-shortest paths, together with
Dijkstra's well known shortest path algorithm. This metho d will b e referred as
the KSP metho d for the rest of this study. Let x
k a
b e the variable representing
the selectionofa'thalternativeforthe k'thsource-destinationpair. LetAb ethe
numb er of alternatives. Let c
k a
b e the cost of selecting a'th alternative for k'th
connection. Let S
ij
b e the set of tuples (k;a). A tuple (k;a) is an element of
set S
ij
if and onlyif a'th alternativefor k'th messagepasses through edge (i;j).
Followingis the formulationfor the KSP metho d.
(IP5) Min P k 2K P 1aA c k a x k a + P (i;j)2E c ij f ij P 1aA x k a = 1 8k 2K P (k ;a)2S ij x k a jWj*f ij 8(i;j)2E x k a 2B f ij 2I
Numb erof variables: (jKjA)+jEj
Numb erof constraints: jKj+jEj
Notice that as A grows large, the formulation is equivalent to ow
formulation,attheexp enseofmoretimeandmemoryrequirementforthebranch
& b ound. It is well known that the p erformance of branch & b ound can b e
improved if a go o d solution is used for pruning. Thus, KSP metho d can b e
KSP metho d.
Procedure KSP
1. For all source-destinationpairs, nd the k shortest paths connecting these
pairs;
2. Use shortest path routing and determinethe b ercost;
3. For a:=2to A
Solve IP5 for a alternatives using the optimum solution for a 1
alternativesas the initial solution of the branch& b ound tree.
Finding a worst case upp er b ound expression for KSP metho d has b een
attempted by the author. The following result is necessary to prove the worst
caseb ehaviorofshortest pathrouting. LetSP(k)b ethe b ercostofthe shortest
path connectingthe sourceand destination of k'thconnection.
Proposition 2 Optimal value of the LP relaxation of IP4 is P
k SP(k )
jWj
Proof: First,noticethatreplacingtheequalitysignsofnetworkconstraintswith
`' will not disturb the structure of the problem. Furthermore, this change will
b e useful since wewill deal with the dual inthe following steps. Noticethat the
structure of the problemisas inFigure 3.1. Notethat N1;N2;:::;NK denotes
the network ow constraint blo cks for connections 1;2;:::;jKj corresp ondingly,
BC denotesthebundleconstraintblo ck, andthe restof theconstraintco ecient
matrix iscomp osed of `0's.
Each square corresp onds to the network constraints of a commo dity. The
rectangular blo ck of constraints are the bundle constraints that bind the
commo dities. Each column of ow variables consists of a `1' and a `-1' in the
N1
N2
N3
NK
BC
b
Figure 3.1: Structure of constraint matrixof IP4
rectangular blo ck. Fib er variables are represented only in the rectangular blo ck
with asingle jWjvalue. Right handsideof the network owconstraintsconsists
of one `1' and one `-1' for each commo dity, and the rest of the right hand side
values are zero. All constraints are `' typ e. The structure of the dual of this
formulationis as follows:
N1T
T
T
T
N2
N3
NK
BCT
c
T
Figure 3.2: Structure of Dualof IP4
Note that every variableis nonnegative, each of the constraints is `' typ e, and
right hand side consists of `0's for constraints corresp onding to ow variables
and c
ij
thebundleconstraintsofIP4,withrighthandsidevaluesofc
ij
. Theseconstraints
can b e restatedas:
w i c ij jWj where w i
is the dual variable corresp onding to the bundle constraint of primary
variablef
ij
. Noticethat inevery oneof the rest of the dual constraints, one and
only one nonzero co ecientexists for thedual variablesrepresentingthe bundle
constraints,whichis`-1'. ApplyingFourier-Motzkineliminationondualvariables
representing the bundleconstraints eliminates these variablesand yields aright
hand side vector consisting of ` cij
jWj
's. In other words, assume that we restate
each of the rest of dual constraints by taking the dual variable representing
the bundle constraint to the right hand side. Notice that each dual variable
corresp onding to a bundle constraint constitutes an upp er b ound on a dual
constraint,andthelowermostpartoftherectangularblo ckimp osesupp erb ounds
onthedualvariablescorresp ondingtobundleconstraints. Thus,wecaneliminate
these variables by replacing them with their corresp onding upp er b ounds. The
structureoftheresultingdualproblemisasfollows,whererighthandsideconsists
of cij jWj values:
N1T
T
T
T
c’T
N2
N3
NK
Min P k 2K P (i;j)2E c ij jWj x ijk s.t. P j2N x ijk P j2N x jik = y ik 8i2N;k 2K NC x ijk 0
Sincethebundleconstraintsthatb oundthecommo ditiestogetherareeliminated,
this problem is minimum cost ow for each commo dity, and clearly, every
commo ditywillfollowthe shortest path formitssource to itsdestination, which
has a cost of SP(k )
jWj
for each commo dity k. So the overall cost of the optimum
solution of this LP is: P
k 2K SP(k )
jWj
. Notice that we did not disturb the problem,
but justmo di ed it. Thus,the optimal value of relaxation of IP4is the sameas
the optimalvalue of this LP. 2
Proposition 3 Let Z
s
denote the value of the shortest path routing and Z
denote the value of the optimal solution. then, Z
s Z (1+ maxc ij minc ij jWj 1 jKj jEj)
Proof: We are trying to construct a feasiblesolution to IP4, given the integral
routing data. Inthiscase, totalcostofthe b ers willb eat least P
k SP(k )
jWj
. Notice
that numb erof b ers onan edgecan b eatmost jWj 1
jWj
lessthanthevalueyielded
by thebundle constraintsof IP4. Considering the worst case, assumeeveryedge
is used in the shortest path routing. Then:
Z s P k SP(k ) jWj +( jWj 1 jWj P (i;j)2E c ij )
Since jEjmaxc
ij P (i;j)2E c ij , Z s P k SP(k ) jWj +( jWj 1 jWj maxc ij jEj)
Dividing and multiplyingthe last termby minc
ij , Z s P k SP(k ) jWj +( jWj 1 jWj maxc ij minc ij jEjminc ij )
ij Z s P k SP(k ) jWj +( jWj 1 jWj maxc ij minc ij jEjminSP(k))
Dividing and multiplyingthe last termby jKj,
Z s P k SP(k ) jWj +( jWj 1 jWj maxc ij minc ij jEjminSP(k) jKj jKj ) Since P k SP(k)jKjminSP(k), Z s P k SP(k ) jWj +( jWj 1 jWj maxcij minc ij jEj P k SP(k ) jKj ) Z s P k SP(k ) jWj (1+ jWj 1 jKj maxc ij minc ij jEj)
Finally, by Prop osition 2, P k SP(k ) jWj Z Z s Z (1+ jWj 1 jKj maxc ij mincij jEj) 2
Theworstcase expressiono ers insightsab outthecostb ehaviorofshortest
path routing. First,ifjWj=1,thenrouting alltheconnectionson their shortest
paths to their destination is the optimal solution, which is obvious. Next, it
impliesthat as the size of the network grows, as the prop ortion of length of the
longest link to the length of the shortest link increases, and as the numb er of
wavelengths available increases the cost b ehavior may not b e very go o d. Last
and the mostimp ortantimplicationis that,as jKj grows large, thep erformance
of shortest path routing improves. Obviously, for large connection request sets
(150 or more), KSP metho d b ecomes harder to apply b ecause of the increasing
numb er of variables. But we know that, as the cardinality of the connection
set increases, shortest-path routing tends to b ehave b etter. Thus, value of A
(numb erof shortest path alternatives)may b e decreased without a great loss of
p erformance when jKj is large.
3.2 Wavelength Assignment and Interchanger
Location Problem
The name of this section suggests two subproblems, but actually,
Wavelength Assignment and Interchanger Lo cationproblems requireinteracting
decisions. A connection may (or may not) change its wavelength assignmentin
a no de if the no de is aWIXC (or not). If interchanger lo cations are xed,then,
each connection can b e broken into several pieces at every WIXC on its route
and assignedseparate wavelengthsforeachpiece. However,the aimofthis study
is to determine the numb erof WIXC's and where they should b e placed. Thus,
these problems are inseparable for our study.
Once routing is xed,the problem b ecomes assigning wavelengths to each
connection at each link it uses. Note that if eachedge consists of a single b er,
and allno des are WSXC's,the problemis equivalentto the graph k-colorability
problem (with k=jWj), whichisknown to b e NP-Complete([2]).
Since three IP formulations involving wavelength assignment and
inter-changer lo cation have b een presented in Chapter 2, using the formulations to
solve the wavelength assignmentand interchanger lo cation problem while xing
therouting,seemedappropriate. IP3wasselectedforthereductioninthenumb er
of variables it o ers. Following is the mo di ed formulation for the case where
formulation is xed. Let r
ijk
b e the routing data, i.e., the amount of ow of
commo dityk from no de i to no de j.
(IP3') Min P (i;j)2E c ij f ij +HC( P i2N a i ) s.t.
w 2W x ijk w = r ijk 8(i;j)2E;k 2K 0 P k 2K (x ijk w +x jik w ) f ij 8(i;j)2E;w2W P j x ijk w P j x jik w M*a i 8w 2W;i2N;k 2K ;w her eY ik 0 x ijk w 2I f ij 2I a i 2B
Numb erof variables: (2jEjjK 0
jjWj)+jEj+jNj
Numb erof constraints: (jEjjK 0
j)+(jEjjWj)+(jK 0
jjN 1jjWj)
Needlessto say,manyvariableswereeliminatedfromthemo delsinceedges
whichdo not carry ows are discarded. Thismo delislaterused inChapter 4for
determiningthe numb erand lo cation of wavelengthinterchangers.
One other formulationto b e presented is for wavelength assignment when
routing is xedand thereare no wavelengthconverters. This formulationisused
forthe purp oseofdeterminingthemaximumnumb erofwavelengthconvertersto
b e placed. Letc
1
b etheoptimal b ercostofrouting allthe connectionswhenall
no des are wavelengthinterchangers. Let c
2
b e the optimal b er cost when none
of the no des are wavelengthinterchangers. Clearly,c
2 c
1
. Let HC denote the
cost of a single wavelength interchanger. If c
2 c
1
HC , then no wavelength
interchangersare requiredand c
2
isthe optimalvalueofthe overallproblem. So,
if aquick way of nding asolution for the no interchanger case is found, it may
b e useful for assessing the value of a wavelength interchanger. Following is the
formulation for wavelengthassignment when routing is xed. Let x
k w
denote if
k'thmessageisassignedwavelengthwor not. LetR
ij
denotethe setof messages
that pass through link (i;j).
(IP6) Min P (i;j)2E c ij f ij s.t.
w 2W x k w = 1 k 2K P k 2Rij x k w f ij 8(i;j)2E;w2W x k w 2B f ij 2I
Numb erof variables: (jKjjWj)+jEj
Numb erof constraints: jKj+(jEjjWj)
3.3 Generating strong lower bounds
During the analysis of the integer multicommo dity ow problem with
variable upp er b ounds, it was noticed that even if the integrality constraints
of the ow variables are dropp ed, the optimal value of the resulting problem
is still close to that of the original problem. Hence, it seemed appropriate
to use an aggregated formulation for multicommo dity ow with variable upp er
b ounds andforceintegralityconstraintsonlyon variablesrepresentingnumb erof
b ers, to obtain a strong lower b ound. Furthermore, this formulation could b e
strengthenedbyaddingcutsetinequalities. ForNFSNETtop ology,lowerb ounds
generated provedto b e equal to the value of the optimal solution of the integer
multicommo dity ow problem with variable upp er b ounds, 88.57% of the time
(93outof105instances). Whenthelowerb oundwasnotequaltothevalueofthe
optimalsolutionofthe integermulticommo dity owproblemwithvariableupp er
b ounds, the di erenceb etweenthe objectivevalues was not morethat5% of the
lower b ound. For larger top ologies, solving the original problem to optimality
was computationally exp ensive, so such data is not available for ARPA2 and
MESH32 top ologies.
In order to determine the cutset inequalities to b e added the following
metho dology was used. Connected minimal subsets of no des for non-planar
top ology (NFSNET), and minimal faces of planar network top ologies (ARPA2
and MESH32) were used. A face is de ned as the remaining connected
Large numb er of subsets of no de set N, led the author to nd a b etter
way of searching for connected S-T partitions. Instead of taking subsets of the
wholeno de setN,itseemedappropriatetoaggregate someno des todecreasethe
total numb erof subsets to b e examined. Aggregated no des werethen connected
with edges to denote if they are connected or not, and a condensed graph was
constructed. This metho d reduced the 14 no de NFSNET top ology to a 7 no de
condensed graph, 21 elementno de ARPA2top ology to 6no de condensedgraph,
and 32 element no de MESH32 top ology to 19 no de condensed graph. Cutset
inequalities generated using the constructed graphs mentioned ab ove proved to
A solution method
4.1 Declaration of the overall procedure
Analysis of the subproblems declared in Chapter 3 led us to the idea of
solving a subproblem, and then solving the rest of the problem while xing the
part solved b efore. Most appropriate choice seemed to rst nd the routing
without wavelength assignment, due to the fact that KSP metho d proved to
b e a very ecient heuristic. When routing is xed, the resulting problem is
the wavelength assignment and interchanger lo cation problem. This problem,
to o, involved a large numb er of variables. Instead of trying to solve the
wavelength assignment and interchanger lo cation problem, rst, a solution for
the casewithoutanywavelengthinterchangersisgenerated. Incaseawavelength
assignmentwithoutanyconvertersandwiththesame b ercostastheroutingcan
b e found, the result of the wavelengthassignmentis optimal. If sucha solution
cannot b e found, next step is to x the numb er of b ers on every edge and to
determine the minimumnumb er of wavelength interchangers and their lo cation
that allows afeasiblewavelengthassignment. Followingistheformaldescription
1. Use KSP metho dforasuitablevalueofA (numb erof alternatives),to nd
a feasiblerouting.
2. SolveIP6 with the routing data fromstep 1
iftheoptimalvalueofIP6isequalto thatofKSPmetho d,stop,existing
solution is optimal
else,go to step 3
3. SolveIP3 with the routing data fromstep 1, the f
ij
values xed to that of
the b estsolution of KSP metho d and HC =1.
4.2 Remarks about the procedure
Obviously, the complexity of the overall pro cedure is exp onential. The
pro ceduretriestosolvelargeIP'sforproblemsknownto b eNP-Hard,atallthree
stages. Fortunately, IP's used for the subproblems tend to b ehave well. Stages
2 and 3 seldom resulted in provably optimal solutions, but they pro duced go o d
solutionsinarelativelyshorttime. Detailedresultsareprovidedattheendofthis
chapter. One imp ortant note is ab out the costingof the alternativesin stage 1.
In termsof the problem,the path of aconnection request do es not matter since
the only cost objects are b ers and wavelength interchangers. However, once
routing is xed, wavelength assignment and consequently nal b er quantities
and wavelength interchanger lo cations are e ected by the choice of path. It was
noticed that many alternative optimal solutions exist for the rst stage. The
author's intuition suggested that as a connection uses more b ers, it interacts
with moreand moreconnections, makingitharderto assignwavelengths. Thus,
each alternative was assigned a small cost, namely 1
1000
'th of the total numb er
of b ers it uses. This way, the formulationtried to minimizethe total b er cost
The metho d prop osed in Section 4.1 was applied to three di erent
top ologies, namely, NFSNET, ARPA2, and MESH32. Randomly generated
source-destination pairs were used during the exp erimentation. Cost of a b er
was assumed to b e equal to its approximate length. 21 data sets for NFSNET,
21 datasets for ARPA2and 9data sets for MESH32 weretested. Atotal of 237
runs haveb eenmadetoobtainthesolutions. Samenumb erofrunswererequired
for obtaining the lowerb ounds. The computerused for the exp erimentationisa
Sun Enterprise4000 withaCPUclo ckof248 Mhzand1024 MBofreal memory.
The co de was develop ed in C, and the callable library of CPLEX 5.0 was used
for mixed integer optimization.
Rest of this chapter is comp osed of gures and detailed tables. Note that
`DS' is the column for the name of the data set, and `NC' is the column that
denotes the numb er of connections a data set involves. To b etter analyze the
e ectsof tracdensityandnumb erofavailablewavelengthson thep erformance
ofthesolutionpro cedure,thepro cedurewasappliedtosetsofconnectionrequests
for di erent numb er of wavelengths available. For example, the pro cedure was
appliedto dataset`ds7'for the NFSNETtop ology, involving40 connections,for
jWj=8;10;12;14;16 where jWj is the numb erof wavelengthsavailable.
Tables do not include time data since the time for the overall pro cedure
was limited to 12 hours of computer time (4 hours for each stage). Esp ecially
stages 2 and 3 tended to nd the b est solution in the early stages of branch &
b ound tree, but sp ent to o much timeto proveoptimality. Since network design
problemsare notto b erep eateddailyinreallife,lengthofthe computationtime
canb ea orded. Lowerb oundcomputationsforNFSNETandARPA2top ologies
yielded the results in a few hours of computer time, whereas computations for
the MESH32 top ology to ok a few days.
The deviations do not exhibit any precise pattern. While they are quite
close to the lower b ounds, there seems no absolute guideline for predicting the
b ehaviour of the results. Empiricalevidencesuggests that the typ e and the size
of thetop ology havethegreatest e ectsonthe results. NextcomesjWjandjKj.
It can b e said that it is harderto obtaina solution with objectivevalue close to
