Table Of ContentTexts in Computer Science
Rodney G. Downey
Michael R. Fellows
Fundamentals
of Parameterized
Complexity
Texts in Computer Science
Editors
DavidGries
FredB.Schneider
Forfurthervolumes:
www.springer.com/series/3191
Rodney G. Downey (cid:2) Michael R. Fellows
Fundamentals
of Parameterized
Complexity
RodneyG.Downey MichaelR.Fellows
SchoolofMathematics,Statisticsand SchoolofEngineeringandInformation
OperationsResearch Technology
VictoriaUniversity CharlesDarwinUniversity
Wellington,NewZealand Darwin,NorthernTerritory,Australia
SeriesEditors
DavidGries FredB.Schneider
DepartmentofComputerScience DepartmentofComputerScience
CornellUniversity CornellUniversity
Ithaca,NY,USA Ithaca,NY,USA
ISSN1868-0941 ISSN1868-095X(electronic)
TextsinComputerScience
ISBN978-1-4471-5558-4 ISBN978-1-4471-5559-1(eBook)
DOI10.1007/978-1-4471-5559-1
SpringerLondonHeidelbergNewYorkDordrecht
LibraryofCongressControlNumber:2013955593
©Springer-VerlagLondon2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub-
lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany
errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect
tothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Downeydedicatesthisbooktohiswife
Kristin,andFellowstohiswifeFrances.
Preface
Parameterizedcomplexity/multivariatecomplexityalgorithmicsisanexcitingfield
ofmodernalgorithmdesignandanalysis,withabroadrangeoftheoreticalandprac-
ticalaspectsthatanswersthevitalneedforefficientalgorithmsbyalmosteveryfacet
ofmodernsociety.Thelastdecadeandahalfhasseenremarkableprogress.There
arenowwholeconferencesdevotedeventosubareasofthefield.Therehasbeenan
exhilarating development of techniques where we see an extended discourse with
a combinatorial problem via parameters as articulated in the original monograph,
DowneyandFellows[247].Wehaveseenthedevelopmentofdeeptechniquesfor
systematicallyattackingthetractabilityofproblems,aswellastechniquesforshow-
ing that the techniques are nearly optimal. This “yin and yang” of parameterized
complexityisastunningendorsementofthemethodology,althoughmanyexcellent
questions remain, not the least of which is why things like SAT-solvers work so
well,althoughtheparameterizedframeworkisprovidinganswersthroughworkby
GaspersandSzeider[340]andothers.
In the preface to our original book, we described how the project began with
someconcretepuzzlementsthatseemedelemental.Inparticular,thegraphminors
projectofRobertsonandSeymourhadsuggestedthenotion,centraltothisfield,of
fixed-parametertractability.
Atthattime,weknewthatwhatisseeminglythemostimportantproblemincom-
binatorialoptimization,theVERTEXCOVERproblem,isfixed-parametertractable,
parameterized in the natural way, by solution size. We knew that the similarly
defined graph optimization problems INDEPENDENT SET and DOMINATING SET
seemedtoresistthepowerofwell-quasiordering.Andweknewareductioncarry-
ingtheissueoffixed-parametertractabilityfromthe INDEPENDENT SET problem
totheDOMINATING SETproblem,butnotviceversa.Itseemedlikeaninteresting
theory should be possible! We had no clear idea if it would be useful. Fired by a
commonloveofsurfingandwine,weembarked!
Afterlegendaryadventures,numerouspapers,andthesacrificingofmanygrad-
uatestudentstowhatwaslongconsideredacult,especiallyinNorthAmerica(are-
ception to our ideas that we did not anticipate), we amassed enough interesting
resultstoofferthefirstbook,conceivedin1990andfinallypublishedin1999.The
firstbookwasfullofbugs(wehopethisonehasfewer),butthatdoesnotseemto
havematteredsomuch,asithadfreshideasandconceptualapproaches,andanice
listofchallengesattheend.Sodoesthisone.
vii
viii Preface
Thefieldofparameterized/multivariatealgorithmsandcomplexityisnowfirmly,
andpermanently,avibrantpartoftheoreticalcomputerscience. Thereisathriving
conferenceseries,hundredsofpapers,twofurtherbooks,1 andanynumberofnew
fieldsofapplications.Thesubjectisevolvingveryrapidlyasbrilliantyoungauthors
remakeitintheirownimage.Inwritingthisbookwewereespeciallykeentoshow-
case the wealth of recent techniques for proving parameterized tractability and to
showcasethepowerfulnewlowerboundtechniques.
Themultivariateperspectivehasproveduseful,evenarguablyessential,tomod-
ernscience,startingwithbioinformaticsasacharterareaforapplications.Twovol-
umes of the Computer Journal [250] essay just some of the areas of applications
ofmultivariatealgorithms.Aswementioninthisbook,thisworkremainshand-in-
handwithalgorithmengineering.
It has become clear that, far from being an accidental incitement to the central
notion of fixed-parameter tractability, deep mathematical structure theory, such as
representedbythegraphminorsproject,isafundamentalcompanionofthecentral
complexitynotion.
Whenthisbookwasbeingprepared,thefirstauthorgaveaseriesoftutorialsat
aCorunafortheconferenceLATA2012.Someoneintheaudienceaskedwhether
masteryofthis(upcoming)bookwouldsufficetounderstandcomplexityissuesin
theirownareaofresearch.Theanswerisdefinitely“Yes”.Masteringareasonable
fragmentofthisbookwillenablearesearchertousethepositiveandnegativetoolk-
itsintheirownresearch.Themultivariateframeworkallowsanextendeddialogwith
aproblem,whichwedescribeinthisbook.
This book is targeted at the beginning graduate student, and accessible to an
advancedsenior(i.e.finalyearundergraduatestudent).Itisalsoaimedatthegeneral
computer scientist, and the mathematically aware scientist seeking tools for their
research.Wehavetriedtomakethematerialasself-containedaspossible.
For many problems we have given a large number of methods for algorithmic
solutionsforthesameproblem.Similarlyforthesameproblemwehaveoftengiven
a series of lower bounds based on ever stronger complexity hypotheses, showing
howthemethodsandideashaveevolvedoverthepast25years.
Wehaveenjoyedteachingmanyyoungresearcherstosurf,andtrottingtheideas
allovertheglobe.Wewelcomeyoutoenjoythefar-reachingideasofmultivariate
algorithmics,theheartbeatofalgorithmsandcomplexity.
Wellington,NewZealand RodneyG.Downey
Darwin,Australia MichaelR.Fellows
1NamelyFlumandGrohe[312]andNiedermeier[546].
Acknowledgements
FirstthankstoSpringerforbeingenthusiasticaboutthisproject,andtoNeeldhara
Misraforbeingsopersistentwithherrequestsforanewbook.
AsalwaysthestaffatSpringerwereprofessional,helpful,andtolerant.Special
thankstoRonanNugent,SimonRees,andRajivMonsurate.
Thisprojectwouldnothavehappenedwithouttheexceptionalsupportprovided
bytheMarsdenFundofNewZealand,andtothewonderfulTuringCentenaryPro-
gramme of 2012 at the Newton Institute where Downey spent the first six months
of 2012. Fellows thanks the Australian Research Council, and the Alexander von
Humboldt Foundation for providing a marvelous year in Germany as a Humboldt
ResearchAwardee.
Special thanks go to those workers in the parameterized world who provided
insight into their work, provided corrections and offered corrections, suggestions,
and exercises. There are many, but special thanks go to Hans Bodlaender, Jianer
Chen,AndrewDrucker,MartinGrohe,MikeLangston,DanielLokshtanov,Stephan
Kreutzer, Dániel Marx, Neeldhara Misra, Frances Rosamond, Saket Saurabh, and
DimitriosThilikos.
We drew on many sources while writing this book, most especially of course
ontheinsightsandresearchoftheparameterized/multivariatecommunity.Helpful
tracking of recent developments in parameterized complexity is provided by Bart
JansenwhomaintainsalistofFPTPapersinConferences,andalistofFPTpapers
onarXivontheparameterizedcomplexitywikiwww.fpt.wikidot.com.Conferences
andseminars,newresults,awards,andprizesareannouncedintheParameterized
ComplexityNewslettereditedbyFrancesRosamond,andarchivedatthewiki.
WethankCathyYoungandGabrielleSchubertfortheirwonderfuleditorialsup-
port.ThankstoJudithEgan,RalphBottesch,andWolfgangMerkleforbeingmaster
spottersoferrors,andthankstoFrancesRosamondforexperthelpwiththegalleys.
ix
Introduction
The world is a highly structured place and the data upon which we wish to run
algorithmsisusuallyhighlystructuredasaconsequence.Virtuallyallarenasofap-
plicationsofalgorithmicsarerepletewithparameters.Theseparametersareaspects
of size, topology, shape, logical depth and the like and reflect this structure. Mul-
tivariatealgorithmics/parameterizedcomplexitysystematicallyseekstounderstand
thecontributionofsuchparameterstotheoverallcomplexityoftheproblem.Asthis
processofdeconstructionrunsitscourse,eachproblemgeneratesitsownextended
dialogwithmultivariatealgorithmics.Theintellectualframeworkthatismultivari-
atealgorithmicshasdeliveredaremarkablyrichsetofdistinctivetoolsforalgorithm
designandanalysis.Itdeliverstoolsforhowtoattackandanalyzeaproblem.More
techniquesarebeingdevelopedeachday.Amazingprogresshasbeenmadeinthe
last 20 years of multivariate research. We have even reached the point where we
havealmostalignmentofupperandlowerboundsofalgorithmics.
Thegoalofthepresentbookistogive,asbestwecan,anaccessibleaccountof
these developments. This gives a snapshot of the current state of the art of multi-
variatealgorithmicsandcomplexity.Itisourbeliefthatalgorithmicsaredeveloped
forrealapplications.Becauseofthis,ourfundamentalbeliefisthefollowing.
Thefutureofalgorithmicsismultivariate.
The reader who is new to this thought, and to the big picture ideas of parame-
terizedcomplexity,shouldbeginbyreadingtheconcreteandhopefullyentertaining
introductiontothefirstbook[247]fromwhichthisbookhasgrown.Thatintroduc-
tion, and many surveys, two other books (Niedermeier [546], Flum-Grohe [312])
andtwoissuesofTheComputerJournal[250]exploretheideasunderpinningthe
area,andforourpurposes,wewillnotdealwithmotivatingtheareanortheranges
ofparametersorapplicationsinthisintroduction.
Sincethisbookisafollowupto[247],inthisintroduction,weturntotheques-
tion:
Whathavewelearnedsincethen?
Wehavelearnedalotabouttheparameterized/multivariateperspectiveonalgo-
rithmsandcomplexitysince1999!
xi
xii Introduction
The multivariate revolution in algorithmics is as much about intellectual
workflowasitisaboutmathematics.
Being a new science, Computer Science is largely misunderstood. High school
studentsoftenhavetheimpressionthatitisanerdything,havingtodowithlearning
tousevarioussoftwarepackages.Whymajorinsomethingsoboring?
ChemistsareamazedthatPh.D.’sareawardedin“programming”(thinkFortran,
evenuntilnow).Biologiststhinkthatifwehaveworkedoutacombinatorialmodel
forthecomputationalproblemofdeducinggeneregulatorynetworksinthepresence
ofsomenoiseintheprimarydata,thenextstepisto“programit”,whichtheyregard
as“routine”.
Prettywelleverybodyoutsideoftheareaofcomputersciencethinksthatifyour
programisrunningtooslowly,whatyouneedisafastermachine.
ButatthebeatingheartofComputerScienceisalgorithmsandcomplexity.This
book is about that beating heart: how mathematical ideas are deployed to design
useful algorithms—for everyone! Google, for example,is essentially an algorithm
tofindandrankwebpages,basedonsurprisinglysophisticatedmathematicalideas.
This book is fundamentally about a multivariate mathematical approach to the
beatingheartofComputerScience:algorithmsandcomplexity.
Physicistsareperhapsourmostcloselyrelatedfamilyofscientists.Ifyoutolda
physicistthatuntilrecently,1 thebeatingmathematicalheartofComputerScience,
thesubjectofalgorithmsandcomplexity,wasdominatedbyaone-dimensionalthe-
oreticalframework(theonedimensionbeingthenumberofbitsinavalidinputfile),
the most likely response would be, “You must be joking. That cannot possibly be
true!”
Yet,ithasbeenso—untilthesubjectofthisbook,anditspredecessor.So,how
didthathappen?
In Physics, there is a generally healthy dynamic (taking the long view, as one
can)betweenTheoreticalPhysicsandExperimentalPhysics.InComputerScience
therehasbeennothinganalogous.
The multivariate point of view of the beating heart of the subject, algorithms
and complexity, has its compass firmly pointed in the direction of an interaction
between the Theoretical and Applied branches of Computer Science, much more
likeourelderintellectualfamilyinPhysics,andthatdirectionisemphasizedinthis
book.
Sowheredidtheone-dimensionalframeworkinTheoreticalComputerScience
come from? It is largely an historical accident. In the 1930’s (before there were
computers)therebeganabranchofpuremathematicsconcernedwiththestudyof
computableandnoncomputablefunctionscalledrecursiontheory.
Amemorablequoteofthefirstauthor(Downey)atanimportantmeetinginLei-
denin2010is:
1Whenthefirstbookwaspublishedin1999,thereisnoquestion,itwasdominatedbytheone-
dimensionaloutlookofPvs.NP.
