The Power of Data Reduction

312 Pages·2013·3.19 MB·English

The Power of Data Reduction
Kernels for Fundamental Graph Problems
Bart M. P. Jansen

The Power of Data Reduction
Kernels for Fundamental Graph Problems
Bart Maarten Paul Jansen
The Power of Data Reduction: Kernels for Fundamental Graph Problems
Bart M. P. Jansen
ISBN 978-90-393-5966-2
9 789039 359662
The cover photo “Matryoshka dolls from Eastern Europe, in intricate handpainted
detail” is (cid:13)c 2010 Hilary Ostapovitch (Crossroads Foundation). It was made
available under a Creative Commons Attribution 2.0 Generic license.
(cid:13)c Bart M. P. Jansen, Utrecht 2012–2013
[email protected]
The Power of Data Reduction
Kernels for Fundamental Graph Problems
De Kracht van Gegevensreductie
De Kern van Fundamentele Graafproblemen
(met een samenvatting in het Nederlands)
Proefschrift
terverkrijgingvandegraadvandoctoraandeUni-
versiteitUtrechtopgezagvanderectormagnificus,
prof. dr. G.J. van der Zwaan, ingevolge het besluit
van het college voor promoties in het openbaar te
verdedigen op maandag 1 juli 2013 des middags te
12.45 uur
door
Bart Maarten Paul Jansen
geboren op 5 augustus 1986
te Nijmegen
Promotor: Prof. dr. J. van Leeuwen
Co-promotor: Dr. H.L. Bodlaender
This research was supported by the Netherlands Organization for Scientific Research
(NWO) through the project “KERNELS: Combinatorial Analysis of Data Reduction”.
Mathematical talent is probably congenital, but aside from that the most
important attribute of a genuine professional mathematician is scholarship. The
scholar is always studying, always ready and eager to learn. The scholar knows
the connections of this specialty with the subject as a whole; he knows not only
the technical details of his specialty, but its history and its present standing; he
knows about the others who are working on it and how far they have reached. He
knowstheliterature, andhetrustsnobody; hehimselfexaminestheoriginalpaper.
He acquires firsthand knowledge not only of its intellectual content, but also of
the date of the work, the spelling of the author’s name, and the punctuation in
the title; he insists on getting every detail of every reference absolutely straight.
The scholar tries to be as broad as possible. No one can know all of mathematics,
but the scholar can succeed in knowing the outline of it all: what are its parts
and what are their places in the whole? These are the things, some of the things,
that go to make up a pro.
—Paul R. Halmos, 1985
(v)

Contents
I Foundations 1
1 Introduction 3
1.1 Capturing Preprocessing by Kernelization . . . . . . . . . . . . . . 5
1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Published Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Parameterized Complexity 11
2.1 The Main Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 The Positive Toolkit . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 The Negative Toolkit . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 Nondeterminism and Advice . . . . . . . . . . . . . . . . . 17
2.2 The Complexity Ecology of Parameters . . . . . . . . . . . . . . . 20
2.2.1 Formalizing Structural Parameterizations . . . . . . . . . . 21
2.2.2 A Table View of Ecology: Problems vs. Parameters. . . . . 26
2.3 A Hierarchy of Parameters. . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Towards the Boundaries of Tractability . . . . . . . . . . . . . . . 30
2.5 Explaining Preprocessing Through the Ecology Program . . . . . . 31
II Tools for Kernelization 33
3 Kernelization Lower Bounds 35
3.1 A Brief History of Kernel Lower Bound Techniques . . . . . . . . . 36
3.2 Polynomial-Parameter Transformations . . . . . . . . . . . . . . . 37
3.3 Cross-Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.2 How Cross-Compositions Yield Lower Bounds . . . . . . . . 40
3.4 Applications of Cross-Composition . . . . . . . . . . . . . . . . . . 42
3.4.1 Clique Parameterized by Vertex Cover . . . . . . . . . . . . 43
3.4.2 Feedback Vertex Set and Odd Cycle Transversal . . . . . . 48
3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 51
(vii)
4 Meta-Theorems for Parameterizations by Vertex Cover 53
4.1 Capturing Polynomial Kernelizability for Parameterizations by Ver-
tex Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Minor Models in Graphs with Small Vertex Covers . . . . . . . . . 56
4.3 Characterization by Few Adjacencies . . . . . . . . . . . . . . . . . 57
4.4 Kernelization for Vertex-Deletion Problems . . . . . . . . . . . . . 61
4.5 Kernelization for Largest Induced Subgraph Problems . . . . . . . 64
4.6 Kernelization for Graph Partitioning Problems . . . . . . . . . . . 67
4.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 71
III Case Studies in the Parameter Hierarchy 73
5 Vertex Cover 75
5.1 Advances in Kernelizing Vertex Cover . . . . . . . . . . . . . . . . 76
5.1.1 Interplay Between Parameterizations . . . . . . . . . . . . . 76
5.1.2 Complexity Boundaries for Vertex Cover . . . . . . . . . . . 79
5.2 Cubic Kernel for Vertex Cover Parameterized by Feedback Vertex Set 81
5.2.1 Independent Sets, Forests, and Matchings . . . . . . . . . . 82
5.2.2 Reducing to a Forest with a Perfect Matching . . . . . . . . 82
5.2.3 Reduction Rules for Clean Instances . . . . . . . . . . . . . 84
5.2.4 Structure of Reduced Instances . . . . . . . . . . . . . . . . 93
5.2.5 Packing Conflict Structures . . . . . . . . . . . . . . . . . . 96
5.2.6 The Kernelization Algorithm . . . . . . . . . . . . . . . . . 105
5.2.7 Discussion of the Kernel . . . . . . . . . . . . . . . . . . . . 108
5.3 Kernel Lower Bound for Parameterization by Outerplanar Modulator109
5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Treewidth 115
6.1 Rigorous Preprocessing for Treewidth Computations . . . . . . . . 116
6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2.1 Properties of Tree Decompositions . . . . . . . . . . . . . . 119
6.2.2 Alternative Characterizations of Treewidth . . . . . . . . . 120
6.3 Cubic Kernel for Treewidth Parameterized by Vertex Cover . . . . 121
6.4 Quartic Kernel for Treewidth Parameterized by Feedback Vertex Set123
6.4.1 Almost Simplicial Vertices . . . . . . . . . . . . . . . . . . . 124
6.4.2 Clique-seeing Paths . . . . . . . . . . . . . . . . . . . . . . 127
6.4.3 A Cut-off Rule . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.4.4 The Kernelization Algorithm . . . . . . . . . . . . . . . . . 136
6.5 Kernel Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.5.1 KernelLowerBoundforTreewidthParameterizedbyClique
Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.5.2 Kernel Lower Bound for Weighted Treewidth Parameterized
by Vertex Cover . . . . . . . . . . . . . . . . . . . . . . . . 147
(viii)