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Optimization and approximation on systems of geometric objects
van Leeuwen, E.J.
Publication date
2009
Link to publication
Citation for published version (APA):
van Leeuwen, E. J. (2009). Optimization and approximation on systems of geometric objects.
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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 . . . 112.2 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 -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 . . . 43Problems . . . 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 -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
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 . . . 159Problems . . . 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
