Table Of ContentOxfordLectureSeriesin
MathematicsanditsApplications31
SeriesEditors
JohnBall DominicWelsh
OXFORD LECTURE SERIES
IN MATHEMATICS AND ITS APPLICATIONS
1. J.C.Baez(ed.):Knotsandquantumgravity
2. I.FonsecaandW.Gangbo:Degreetheoryinanalysisandapplications
3. P.L.Lions:Mathematicaltopicsinfluidmechanics,Vol.1:Incompressible
models
4. J.E.Beasley(ed.):Advancesinlinearandintegerprogramming
5. L.W.BeinekeandR.J.Wilson(eds):Graphconnections:Relationships
betweengraphtheoryandotherareasofmathematics
6. I.Anderson:Combinatorialdesignsandtournaments
7. G.DavidandS.W.Semmes:Fracturedfractalsandbrokendreams
8. OliverPretzel:Codesandalgebraiccurves
9. M.KarpinskiandW.Rytter:Fastparallelalgorithmsforgraphmatching
problems
10. P.L.Lions:Mathematicaltopicsinfluidmechanics,Vol.2:Compressible
models
11. W.T.Tutte:GraphtheoryasIhaveknownit
12. AndreaBraidesandAnnelieseDefranceschi:Homogenizationofmultiple
integrals
13. ThierryCazenaveandAlainHaraux:Anintroductiontosemilinearevolution
equations
14. J.Y.Chemin:Perfectincompressiblefluids
15. GiuseppeButtazzo,MarianoGiaquintaandStefanHildebrandt:
One-dimensionalvariationalproblems:anintroduction
16. AlexanderI.BobenkoandRuediSeiler:Discreteintegrablegeometryand
physics
17. DoinaCioranescuandPatriziaDonato:Anintroductiontohomogenization
18. E.J.JansevanRensburg:Thestatisticalmechanicsofinteractingwalks,
polygons,animalsandvesicles
19. S.Kuksin:Hamiltonianpartialdifferentialequations
20. AlbertoBressan:Hyperbolicsystemsofconservationlaws:the
one-dimensionalCauchyproblem
21. B.Perthame:Kineticformulationofconservationlaws
22. A.Braides:Gamma-convergenceforbeginners
23. RobertLeeseandStephenHurley:Methodsandalgorithmsforradiochannel
assignment
24. CharlesSempleandMikeSteel:Phylogenetics
25. LuigiAmbrosioandPaoloTilli:TopicsonAnalysisinMetricSpaces
26. EduardFeireisl:DynamicsofViscousCompressibleFluids
27. AntonínNovotnýandIvanStraškraba:IntroductiontotheMathematical
TheoryofCompressibleFlow
28. PavolHellandJarikNesetril:GraphsandHomomorphisms
29. PavelEtingofandFredericLatour:ThedynamicalYang-Baxterequation,
representationtheory,andquantumintegrablesystems
30. JorgeRamirezAlfonsin:TheDiophantineFrobeniusProblem
31. RolfNiedermeier:InvitationtoFixed-ParameterAlgorithms
Invitation to Fixed-Parameter
Algorithms
Rolf Niedermeier
InstitutfürInformatik,Friedrich-Schiller-UniversitätJena
1
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ISBN 0–19–856607–7 978–0–19–856607–6
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PREFACE
This book grew out of my habilitationsschrift at the University of Tu¨bingen in
2002. Currently, there is only one monograph dealing with the issue of fixed-
parameter algorithms:Rod G. Downey and Michael R. Fellows’groundbreaking
monographParameterizedComplexity (1999).Sincethentherehavebeennumer-
ous new results in this field of exactly solving combinatorially hard problems.
Moreover, Downey and Fellows’ monograph focuses more on structural com-
plexity theory issues than on concrete algorithm design and analysis. By way
of contrast, the objective of this book is to focus on the algorithmic side of
parameterized complexity, giving a fresh view of this highly innovative field of
algorithmic research.
The book is divided into three parts:
1. abroadintroductionthatprovidesthe generalphilosophyandmotivation;
2. apartonalgorithmicmethodsdevelopedovertheyearsinfixed-parameter
algorithmics, forming the core of the book; and
3. a final section discussing the essentials of parameterized hardness theory,
focusingfirstonW[1]-hardness,whichparallelsNP-hardness,thenstating
somerelationstopolynomial-timeapproximationalgorithms,andfinishing
upwithalistofselectedcasestudiestoshowthewiderangeofapplicability
of the methodology presented.
The book is intended for advanced students in computer science and related
fields as well as people generally working with algorithms for discrete problems.
It has particular relevance when studying ways to cope with computational in-
tractability as expressed by NP-hardness theory.
The reader is recommended to start with Part I, but Parts II and III do
not need to be read in the given order. Thus, from Chapter 7 on (with a few
exceptions) there are almost no restrictions concerning the chosen order. The
material presented can be used to form a course exclusively dedicated to the
topicoffixed-parameteralgorithmsaswellastoprovidesupplementarymaterial
for an advanced algorithms class.
We believe that the concept of fixed-parameter tractability is fundamental
forthealgorithmicsofcomputationallyharddiscreteproblems.Duetotheubiq-
uity of the proposed problem parameterization approach discussed here, fixed-
parameter algorithms should be seen as basic knowledge for every algorithm
designer. May this book help to spread this news.
v
ACKNOWLEDGEMENTS
To all who helped, particularly the unnamed ones!
ThefirsthintthatIshouldstudyparameterizedcomplexitycamefromKlaus-
Jo¨rn Lange. He pointed me to strange things like “towers of complexity” or
“parameterizationoflanguagesbytheslice”.Shortlyafterwards,duringmy1998
stay at Charles University, Prague, Jaroslav Neˇsetˇril showed me a bunch of pa-
pers by Rod G. Downey and Michael R. Fellows. This prompted my first steps
in researching fixed-parameter algorithms in enjoyable cooperation with Peter
Rossmanith.
However, I am most grateful to my (partially former) Ph.D. students who
share(d) their ideas and time with me. I list them in alphabetical order:Jochen
Alber,MichaelDom,JiongGuo,JensGramm,FalkHu¨ffner,andSebastianWer-
nicke. They helped me a lot in countless ways and without them this book
would not exist. In addition, I greatly profited from working with my gradu-
ate students Nadja Betzler, Britta Dorn, Frederic Dorn, Erhan Kenar, Hannes
Moser, Amalinda Oertel, David Pricking, Daniel Raible, Marion Renner, Chris-
tian Ro¨delsperger, Ramona Schmid, Anke Truss, and Johannes Uhlmann. All
of them have been infected with fixed-parameter algorithmics in the broadest
sense.
Finally,specialthanksgotoHansBodlaender,MikeFellows,HenningFernau,
Edward Hirsch, and Ton Kloks for collaborations from which I have profited
greatly.
IthankRodDowneyformakingtheconnectionwithOxfordUniversityPress
andthestaffatOxfordUniversityPressforasmoothandenjoyablecooperation.
I apologize for omitting further names here—there are too many to name
them all and it would be too dangerous to forget one of them.
vi
CONTENTS
I FOUNDATIONS
1 Introduction to Fixed-Parameter Algorithms 3
1.1 The satisfiability problem 4
1.2 An example from railway optimization 7
1.3 A communication problem in tree networks 10
1.4 Summary 12
1.5 Exercises 13
1.6 Bibliographical remarks 14
2 Preliminaries and Agreements 17
2.1 Basic sets and problems 17
2.2 Model of computation and running times 17
2.3 Strings and graphs 18
2.4 Complexity and approximation 20
2.5 Bibliographical remarks 21
3 Parameterized Complexity Theory—A Primer 22
3.1 Basic theory 22
3.2 Interpreting fixed-parameter tractability 27
3.3 Exercises 29
3.4 Bibliographical remarks 29
4 Vertex Cover—An Illustrative Example 31
4.1 Parameterizing 32
4.2 Specializing 33
4.3 Generalizing 34
4.4 Counting or enumerating 34
4.5 Lower bounds 35
4.6 Implementing and applying 35
4.7 Using vertex cover structure for other problems 36
4.8 Exercises 38
4.9 Bibliographical remarks 38
5 The Art of Problem Parameterization 41
5.1 Parameter really small? 41
5.2 Guaranteed parameter value? 42
5.3 More than one obvious parameterization? 43
5.4 Close to “trivial” problem instances? 45
5.5 Exercises 47
5.6 Bibliographical remarks 47
vii
viii CONTENTS
6 Summary and Concluding Remarks 49
II ALGORITHMIC METHODS
7 Data Reduction and Problem Kernels 53
7.1 Basic definitions and facts 55
7.2 Maximum Satisfiability 58
7.3 Cluster Editing 60
7.4 Vertex Cover 64
7.4.1 Kernelization based on matching 64
7.4.2 Kernelization based on linear programming 68
7.4.3 Kernelization based on crown structures 69
7.4.4 Comparison and discussion 72
7.5 3-Hitting Set 72
7.6 Dominating Set in Planar Graphs 74
7.6.1 The neighborhood of a single vertex 74
7.6.2 The neighborhood of a pair of vertices 77
7.6.3 Reduced graphs and the problem kernel 79
7.7 On lower bounds for problem kernels 80
7.8 Summary and concluding remarks 82
7.9 Exercises 83
7.10 Bibliographical remarks 85
8 Depth-Bounded Search Trees 88
8.1 Basic definitions and facts 91
8.2 Cluster Editing 93
8.3 Vertex Cover 98
8.4 Hitting Set 101
8.5 Closest String 103
8.6 Dominating Set in Planar Graphs 107
8.6.1 Data reduction rules 108
8.6.2 Main result and some remarks 109
8.7 Interleaving search trees and kernelization 110
8.7.1 Basic methodology 111
8.7.2 Interleaving is necessary 113
8.8 Automated search tree generation and analysis 114
8.9 Summary and concluding remarks 119
8.10 Exercises 120
8.11 Bibliographical remarks 121
9 Dynamic Programming 124
9.1 Basic definitions and facts 125
9.2 Knapsack 126
9.3 Steiner Problem in Graphs 128
9.4 Multicommodity Demand Flow in Trees 131
CONTENTS ix
9.5 Tree-structured variants of Set Cover 136
9.5.1 Basic definitions and facts 136
9.5.2 Algorithm for Path-like Weighted Set Cover 139
9.5.3 Algorithm for Tree-like Weighted Set Cover 140
9.6 Shrinking search trees 145
9.7 Summary and concluding remarks 146
9.8 Exercises 147
9.9 Bibliographical remarks 148
10 Tree Decompositions of Graphs 150
10.1 Basic definitions and facts 151
10.2 On the construction of tree decompositions 153
10.3 Planar graphs 155
10.4 Dynamic programming for Vertex Cover 160
10.5 Dynamic programming for Dominating Set 164
10.6 Monadic second-order logic (MSO) 169
10.7 Related graph width parameters 172
10.8 Summary and concluding remarks 174
10.9 Exercises 175
10.10Bibliographical remarks 176
11 Further Advanced Techniques 177
11.1 Color-coding 178
11.2 Integer linear programming 181
11.3 Iterative compression 184
11.3.1 Vertex Cover 185
11.3.2 Feedback Vertex Set 187
11.4 Greedy localization 190
11.4.1 Set Splitting 191
11.4.2 Set Packing 193
11.5 Graph minor theory 195
11.6 Summary and concluding remarks 197
11.7 Exercises 198
11.8 Bibliographical remarks 199
12 Summary and Concluding Remarks 201
III SOME THEORY, SOME CASE STUDIES
13 Parameterized Complexity Theory 205
13.1 Basic definitions and concepts 206
13.1.1 Parameterizedreducibility 207
13.1.2 Parameterizedcomplexity classes 209
13.2 The complexity class W[1] 212
13.3 Concrete parameterized reductions 216
13.3.1 W[1]-hardness proofs 218
x CONTENTS
13.3.2 Further reductions and W[2]-hardness 226
13.4 Some recent developments 230
13.4.1 Lower bounds and the complexity class M[1] 230
13.4.2 Lower bounds and linear FPT reductions 232
13.4.3 Machine models, limited nondeterminism, and
bounded FPT 233
13.5 Summary and concluding remarks 234
13.6 Exercises 235
13.7 Bibliographical remarks 235
14 Connections to Approximation Algorithms 237
14.1 Approximation helping parameterization 238
14.2 Parameterizationhelping approximation 239
14.3 Further (non-)relations 241
14.4 Discussion and concluding remarks 241
14.5 Bibliographical remarks 242
15 Selected Case Studies 243
15.1 Planar and more general graphs 243
15.1.1 Planar graphs 243
15.1.2 More general graphs 245
15.2 Graph modification problems 245
15.2.1 Graph modification and hereditary properties 246
15.2.2 Feedback Vertex Set revisited 247
15.2.3 Graph Bipartization 248
15.2.4 Minimum Fill-In 249
15.2.5 Closest 3-Leaf Power 250
15.3 Miscellaneous graph problems 251
15.3.1 Capacitated Vertex Cover 251
15.3.2 Constraint Bipartite Vertex Cover 253
15.3.3 Graph Coloring 255
15.3.4 Crossing Number 256
15.3.5 Power Dominating Set 257
15.4 Computational biology problems 258
15.4.1 Minimum Quartet Inconsistency 259
15.4.2 Compatibility of Unrooted Phylogenetic Trees 261
15.4.3 LongestArc-PreservingCommonSubsequences 262
15.4.4 Incomplete Perfect Path Phylogeny Haplotyp-
ing 264
15.5 Logic and related problems 266
15.5.1 Satisfiability 266
15.5.2 Maximum Satisfiability 268
15.5.3 Constraint satisfaction problems 269
15.5.4 Database queries 270
15.6 Miscellaneous problems 271
