Contributions to the Minimum Linear Arrangement Problem

209 Pages·2010·1.62 MB·English

by  Hanna Seitz

INAUGURAL-DISSERTATION
zurErlangungderDoktorwürdeder
Naturwissenschaftlich-MathematischenGesamtfakultät
derRuprecht-Karls-UniversitätHeidelberg
vorgelegtvon
Diplom–MathematikerinHannaSeitz,geborenePeters
ausCastrop-Rauxel
TagdermündlichenPrüfung:.April

Contributions to the
Minimum Linear Arrangement
Problem
Gutachter: Prof.Dr.GerhardReinelt
Prof.Dr.Dr.h.c.HansGeorgBock

Zusammenfassung
DasMinimumLinearArrangementProblem(MinLA)bestehtdarin,füreinengewich-
teten Graphen eine lineare Anordnung der Knoten zu bestimmen, welche die ge-
wichteteSummederKantenlängenminimiert.DievorliegendeArbeituntersuchtden
NutzeneinerneuenModelierungimRahmeneinesBranch-and-Cut-and-PriceAlgo-
rithmuszuroptimalenLösungdesMinLA.DenKernderModellierungbildenbinäre
Variablend ,diegenaudanndenWert1haben,wenndieKnoteniund jinderPer-
ijk
mutationdieDistanzkhaben.WirpräsentierenangepassteFormulierungenfürdicht-
und dünnbesetzte Graphen und erläutern die Realisierung eines Branch-and-Cut-
and-Price Algoritmus’. Desweiteren werden die verschiedenen Varianten des Algo-
rithmus’diskutiertundevaluiert.ZumStudiumdertheoretischenAspektedesMinLA
leistenwireinenBeitragmitderCharakterisierungeinerRelaxierungdeszugehörigen
Polyeders.
Abstract
TheMinimumLinearArrangementproblem(MinLA)consistsinfindinganordering
of the nodes of a weighted graph, such that the sum of the weighted edge lengths is
minimized. We report on the usefulness of a new model within a branch-and-cut-
and-pricealgorithmforsolvingMinLAproblemstooptimality.Thekeyideaistoin-
troducebinaryvariablesd ,thatareequalto1ifnodesiand jhavedistancekinthe
ijk
permutation.Wepresentformulationsforcompleteandforsparsegraphsandexplain
the realization of a branch-and-cut-and-price algorithm. Furthermore, its different
settingsarediscussedandevaluated. Tothestudyofthetheoreticalaspectsconcern-
ingtheMinLA,wecontributeacharacterizationofarelaxationofthecorresponding
polyeder.
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AboveallIwanttothankProf.G.Reineltwhoallowedmetobalanceresearchand
teaching,averyinterestingcombination.Furthermore,hewasmysoundingboard
whenneededandgavemeinvaluableadvice.
ThanksareduealsotoProf.G.Bockforwritingthereportandbeingveryflexiblein
adaptingtothetightschedule.
MarcusOswaldwasthebestteacherimaginableforon–the–job–training.
Furthermore,Iamtrulythankfulforhiscountlesssuggestionsandunfailingclear
explanations.
IamverygreatfultohavehadThorstenBonatoasmycolleague.Hewasalways
willingtolistencarefullyandtirelessinhelpingmetofindanswers.
CaraCocking,MarkusSpethandPeiWangprovidedaveryrelaxedatmospherein
theoffice.Thanksforbeingsuchniceco-workers.
IamtrulythankfulforthementoringofDirkO.Theis.Workingwithhimwas
tremendouslyinspiringandencouraging.Iappreciatedhiscompetenceandfeedback
aswellashisneverendingenthusiasm.Itmadeagreatdifferenceduringallthese
years.
ToCatherineProux–Wielandahuge"thankyou"fortheexcellentmanagementofall
administrativetasks.Hercaringdelightfulnaturecreatesaverypleasantworking
atmosphere.
ThanksalsotoGeorgiosNikolisandAdrianDempwolffforkeepingthecomputer
systemrunning.Theirattentiontoeverydetailmadeworkingapleasure.
WithoutthepresenceofKarinTenschertoursociallifewouldhavebeenmuchless
exciting.Thanksforbeinginterested,givingemotionalsupportaswellassharing
personalperspectives.
OnnumerousoccasionsIexperiencedgreatcooperationbetweenvariousgroups
andmembersoftheinstitute.InparticularIwanttothankProf.K.Ambos–Spies,
Prof.T.LudwigandProf.B.Paechfortheirpersonalengagement.
Eventhoughandmuchtomyregret,wedidnotspendmuchtimetogether,Iwantto
thankProf.E.Fernandez,Barcelona,forinterestingscientificdiscussions.
IthankProf.A.Letchford,Lancaster(UK),forourinspiringdiscussionsand
exceptionallypleasantteamwork.Ithasbeenagreathonortodoresearchwithhim.
FortheproofreadingpartsofmythesisIthankThorstenBonato,ElsbethDuke,
SarahMaidstone,MarcusOswald,RobertSchwarz,ChristophSchwitzer,Markus
Speth,DirkO.Theis,andStefanWiesberg.Nevertheless,Itakefullresponsibilityfor
anyerrorsthatmayremain.
Overalltheseyearsmyfamilyandalotoffriendsinfluencedmeenormouslybytheir
encouragementandsupport.Iamtrulyindebtedtoallminorandmajor
contributions.
LastbutnotleastmyheartfeltthanksgotomyparentsHeikeandFriedhelmPeters
towhomIamendebtedthemost.