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Optimization and Approximation on Systems of Geometric Objects 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 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 -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 -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 Chapter 1 Introduction Can a combinatorial optimization problem be approximated better if it is de- termined by a system of geometric objects? Many combinatorial optimization problems are NP-hard to solve and frequently even hard to approximate. On systems of geometric objects however, these same problems can usually be solved efficiently or approximated well. The computational complexity of such geometric optimization problems highly depends on the underlying system of objects. We consider objects that arise naturally in diverse applications. For instance, the broadcasting range of a wireless device can be seen as a disk, an atom is just a sphere, and a label on a map is a rectangle. In this thesis, we consider hard combinatorial optimization problems on systems of geometric objects, motivated by applications in for example wireless networks, computational biology, and map labeling. 1.1 Optimization Problems and Systems of Geometric Objects Given a combinatorial optimization problem and its objective function, one preferably is able to compute the optimum objective value and a correspond- ing solution in an efficient way. Unfortunately, depending on one's notion of efficient, this is not always possible. The measure we use to decide whether an algorithm is efficient is polynomial running time. That is, whether the algorithm returns the optimum objective value (and possibly a corresponding solution) using a number of `basic steps' that is polynomial in the size of the problem instance. An optimization problem having such an algorithm is called polynomial-time solvable. Many optimization problems are solvable in polynomial time. However, there are also many optimization problems that are not known to be solvable in polynomial time. In fact, there is very strong belief that these problems cannot be solved in polynomial time. To support this idea, a class of hard optimization problems has been identified, namely the class of NP-hard opti- mization problems. The most sensible way to work around NP-hardness while still restrict- ing to polynomial-time algorithms is to try and find approximate solutions to such optimization problems. An approximation algorithm returns an objec- 1 2 Chapter 1. Introduction tive value (and possibly a corresponding solution) within a certain (additive or multiplicative) factor of the optimum objective value. Approximation algo- rithms have been developed for many different optimization problems and are widely used in practice. We should note however that sometimes one can even show that approximating a problem within a certain factor is NP-hard. In this field of hard optimization problems, we try to ascertain the influ- ence on the computational complexity of these problems when using a system of geometric objects as the underlying combinatorial structure. Since many optimization problems use graphs as their underlying structure, it might be no surprise that one of the central notions of this thesis is a graph induced by a system of geometric objects: a geometric intersection graph. Given a system of geometric objects, the vertices of the induced graph correspond to the given objects and there is an edge between two vertices if the corresponding objects intersect. There are many famous examples of geometric intersection graphs, for instance the classic interval graph is an intersection graph of intervals on the real line. For this thesis however, we are concerned with intersection graphs of objects in two- and higher-dimensional space, such as (unit) disk graphs, which are intersection graphs of (equally-sized) disks. Most classical optimization problems are still NP-hard on (intersection graphs of) systems of two-dimensional geometric objects [17, 67, 110, 157, 267]. However, there is considerable evidence (e.g. [4, 68, 74, 103, 150, 154, 201, 279]) to support the claim that approximating these problems is easier, meaning that better approximation algorithms exist, on systems of geometric objects than in general. The goal of this thesis is to further investigate this claim. In this thesis we study the approximability of hard com- binatorial optimization problems on (intersection graphs of) systems of geometric objects. As part of this study, we give both approximation algorithms and inapprox- imability results. In the process, we show that the approximability highly depends on the shape and size of the geometric objects under consideration. 1.2 Application Areas It turns out that optimization problems on systems of geometric objects, and particularly on geometric intersection graphs, occur in many application areas. We consider some of the foremost application areas and associated problems and systems of objects below.
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