Optimization and approximation on systems of geometric objects van Leeuwen, EJ

265 Pages·2017·2.17 MB·English

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Optimization and approximation on systems of geometric objects
van Leeuwen, E.J.
Publication date
2009
Document Version
Final published version
Link to publication
Citation for published version (APA):
van Leeuwen, E. J. (2009). Optimization and approximation on systems of geometric objects.
[Thesis, externally prepared, Universiteit van Amsterdam].
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Download date:15 Feb 2023
Optimization and Approximation
on
Systems of Geometric Objects
Erik Jan van Leeuwen
This research has been carried out at the Centrum Wiskunde & Informatica
in Amsterdam.
Part of this research has been funded by the Dutch BSIK/BRICKS project.
Copyright (cid:13)c 2009 by Erik Jan van Leeuwen.
Printed and bound by Ipskamp Drukkers B.V., The Netherlands.
ISBN 978-90-9024317-7
Optimization and Approximation
on
Systems of Geometric Objects
ACADEMISCH PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Universiteit van Amsterdam
op gezag van de Rector Magnificus
prof. dr. D.C. van den Boom
ten overstaan van een door het college voor promoties
ingestelde commissie,
in het openbaar te verdedigen in de Agnietenkapel
op dinsdag 16 juni 2009, te 10:00 uur
door
Erik Jan van Leeuwen
geboren te Utrecht
Promotiecommissie:
Promotor:
prof. dr. A. Schrijver
Overige leden:
prof. dr. M.T. de Berg
prof. dr. H.M. Buhrman
prof. dr. T. Erlebach
prof. dr. G.J. Woeginger
dr. L. Torenvliet
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Do not disturb my circles!
– last words of Archimedes
(287 BC – 212 BC)

Contents
1 Introduction 1
1.1 Optimization Problems and Systems of Geometric Objects . . . 1
1.2 Application Areas . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Wireless Networks . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Wireless Network Planning . . . . . . . . . . . . . . . . 3
1.2.3 Computational Biology . . . . . . . . . . . . . . . . . . 4
1.2.4 Map Labeling . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.5 Further Applications . . . . . . . . . . . . . . . . . . . . 4
1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Published Papers . . . . . . . . . . . . . . . . . . . . . . 7
I Foundations 9
2 Primer on Optimization and Approximation 11
2.1 Classic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Asymptotic Approximation Schemes . . . . . . . . . . . . . . . 14
3 Guide to Geometric Intersection Graphs 17
3.1 Intersection Graphs. . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.1 Interval Graphs and Generalizations . . . . . . . . . . . 18
3.1.2 Intersection Graphs of Higher Dimensional Objects . . . 20
3.2 Disk Graphs and Ball Graphs . . . . . . . . . . . . . . . . . . . 21
3.2.1 Models for Wireless Networks . . . . . . . . . . . . . . . 23
3.3 Relation to Other Graph Classes . . . . . . . . . . . . . . . . . 25
3.3.1 Relation to Planar Graphs. . . . . . . . . . . . . . . . . 26
4 Geometric Intersection Graphs and Their Representation 29
4.1 Scalable and (cid:15)-Separated Objects . . . . . . . . . . . . . . . . . 29
4.2 Finite Representation . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Polynomial Representation and Separation. . . . . . . . . . . . 34
4.3.1 From Representation to Separation . . . . . . . . . . . . 35
4.3.2 From Separation to Representation . . . . . . . . . . . . 37
II Approximation on Geometric Intersection Graphs 41
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
vii
viii Contents
5 Algorithms on Unit Disk Graph Decompositions 49
5.1 Graph Decompositions . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 Algorithms on Strong, Relaxed Tree Decompositions . . . . . . 54
5.3.1 Maximum Independent Set and Minimum Vertex Cover 55
5.3.2 Minimum Dominating Set . . . . . . . . . . . . . . . . . 56
5.3.3 Minimum Connected Dominating Set . . . . . . . . . . 59
5.4 Unit Disk Graphs of Bounded Thickness . . . . . . . . . . . . . 62
6 Density and Unit Disk Graphs 67
6.1 The Density of Unit Disk Graphs . . . . . . . . . . . . . . . . . 67
6.2 Relation to Thickness . . . . . . . . . . . . . . . . . . . . . . . 68
6.3 Approximation Schemes . . . . . . . . . . . . . . . . . . . . . . 70
6.3.1 Maximum Independent Set . . . . . . . . . . . . . . . . 71
6.3.2 Minimum Vertex Cover . . . . . . . . . . . . . . . . . . 74
6.3.3 Minimum Dominating Set . . . . . . . . . . . . . . . . . 77
6.3.4 Minimum Connected Dominating Set . . . . . . . . . . 79
6.3.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . 82
6.4 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.5 Connected Dominating Set on Graphs Excluding a Minor . . . 88
7 Better Approximation Schemes on Disk Graphs 91
7.1 The Ply of Disk Graphs . . . . . . . . . . . . . . . . . . . . . . 91
7.2 Approximating Minimum Vertex Cover . . . . . . . . . . . . . 92
7.2.1 A Close to Optimal Vertex Cover . . . . . . . . . . . . . 93
7.2.2 Properties of the size- and sol-Functions . . . . . . . . . 93
7.2.3 Computing the size- and sol-Functions . . . . . . . . . . 95
7.2.4 An eptas for Minimum Vertex Cover . . . . . . . . . . . 98
7.3 Approximating Maximum Independent Set . . . . . . . . . . . 99
7.4 Further Improvements . . . . . . . . . . . . . . . . . . . . . . . 104
7.5 Maximum Nr. of Disjoint Unit Disks Intersecting a Unit Square 106
8 Domination on Geometric Intersection Graphs 113
8.1 Small (cid:15)-Nets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.2 Generic Domination . . . . . . . . . . . . . . . . . . . . . . . . 116
8.3 Dominating Set on Geometric Intersection Graphs . . . . . . . 119
8.3.1 Homothetic Convex Polygons . . . . . . . . . . . . . . . 119
8.3.2 Regular Polygons . . . . . . . . . . . . . . . . . . . . . . 123
8.3.3 More General Objects . . . . . . . . . . . . . . . . . . . 125
8.4 Disk Graphs of Bounded Ply . . . . . . . . . . . . . . . . . . . 126
8.4.1 Ply-Dependent Approximation Ratio . . . . . . . . . . . 127
8.4.2 A Constant Approximation Ratio . . . . . . . . . . . . . 129
8.4.3 A Better Constant . . . . . . . . . . . . . . . . . . . . . 138
8.5 Hardness of Approximation . . . . . . . . . . . . . . . . . . . . 145
8.5.1 Intersection Graphs of Polygons . . . . . . . . . . . . . 146
Contents ix
8.5.2 Intersection Graphs of Fat Objects . . . . . . . . . . . . 148
8.5.3 Intersection Graphs of Rectangles . . . . . . . . . . . . 150
III Approximating Geometric Coverage Problems 157
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9 Geometric Set Cover and Unit Squares 165
9.1 A ptas on Unit Squares . . . . . . . . . . . . . . . . . . . . . . 165
9.1.1 Geometric Budgeted Maximum Coverage . . . . . . . . 175
9.1.2 Optimality and Relation to Domination . . . . . . . . . 177
9.2 Hardness of Approximation . . . . . . . . . . . . . . . . . . . . 178
10 Geometric Unique and Membership Coverage Problems 181
10.1 Unique Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . 181
10.1.1 Approximation Algorithm on Unit Disks . . . . . . . . . 182
10.1.2 Budgets and Satisfactions . . . . . . . . . . . . . . . . . 185
10.1.3 Approximation Algorithm on Unit Squares . . . . . . . 187
10.2 Unique Coverage on Disks of Bounded Ply . . . . . . . . . . . . 188
10.2.1 Properties of the cost- and sol-Functions . . . . . . . . . 189
10.2.2 Computing the cost- and sol-Functions . . . . . . . . . . 192
10.2.3 The Approximation Algorithm . . . . . . . . . . . . . . 194
10.3 Geometric Membership Set Cover . . . . . . . . . . . . . . . . . 196
10.4 Hardness of Approximation . . . . . . . . . . . . . . . . . . . . 199
11 Conclusion 207
Bibliography 209
Author Index 237
Index 243
Samenvatting 249
Summary 251
Acknowledgments 253