The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi B.S., Sharif University of Technology, 2000 M.S., University of Waterloo, 2001 Submitted to the Department of Mathematics in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2005 °c MohammadTaghi Hajiaghayi, 2005. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author.............................................................. Department of Mathematics April 29, 2005 Certi¯ed by. Erik D. Demaine Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by . Rodolfo Ruben Rosales Chairman, Applied Mathematics Committee Accepted by . Pavel I. Etingof Chairman, Department Committee on Graduate Students 2 The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi Submitted to the Department of Mathematics on April 29, 2005, in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY Abstract Our newly developing theory of bidimensional graph problems provides general techniques for designing e±cient ¯xed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k £ k grid graph (and similar graphs) grows with k, typically as ­(k2), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph embeddable in a surface of bounded genus has treewidth bounded above by the square root of the problem's solution value. These properties lead to e±cient|often subexponential|¯xed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes. One type of minor-closed graph class of particular relevance has bounded local treewidth, in the sense that the treewidth of a graph is bounded above in terms of the diameter; indeed, we show that such a bound is always at most linear. The bidimensionality theory uni¯es and improves several previous results. The theory is based on algorithmic and combinatorial extensions to parts of the Robertson-Seymour Graph Minor Theory, in particular initiating a parallel theory of graph contractions. The foundation of this work is the topological theory of drawings of graphs on surfaces and our results regarding the relation (the linearity) of the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more generally in graphs excluding a ¯xed graph H as a minor. In this thesis, we also develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L1 (and even Euclidean embeddings) are insu±cient, but that the additional structure provided by many embedding theorems does su±ce for our purposes. p We obtain an O( log n) approximation for min-ratio vertex cuts in general graphs, based on a new semide¯nite relaxation of the problem, and a tight analysis of the integrality gap p which is shown to be £( log n). We also prove various approximate max-flow/min-vertex- 3 cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n). These p results have a number of applications. We exhibit an O( log n) pseudo-approximation for ¯nding balanced vertex separators in general graphs. Furthermore, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to ¯nd a tree p decomposition of width at most O(k log k), whereas previous algorithms yielded O(k log k). For graphs excluding a ¯xed graph as a minor, we give a constant-factor approximation for the treewidth; this via the bidimensionality theory can be used to obtain the ¯rst polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs. Thesis Supervisor: Erik D. Demaine Title: Associate Professor of Electrical Engineering and Computer Science 4 Acknowledgments First and foremost, I am deeply indebted to my thesis advisor, Professor Erik De- maine, and my academic advisor, Professor Tom Leighton, for being enthusiastic and genius supervisors. Erik and I were more friends than advisor and advisee and we spent countless hours together exchanging puzzles, research ideas, and philosophies. Between us, we have explored lots of plausible ideas in algorithm design, many two or more times. His insights (technical and otherwise) have been invaluable to me. Erik also helped me a lot with his useful suggestions on the presentation of the re- sults and the writing style. I am grateful to Tom for his perceptiveness, and his deep insights over the years on my research. He provided me a great source of inspiration. Both Erik and Tom have been extremely generous in giving me a great deal of their valuable time and sharing with me their ideas and insights. This research would have been impossible without their helps and encouragements. I thank Professor Uriel Feige, Professor Fedor Fomin, Professor Naomi Nishimura, Professor Prabhakar Ragde, Professor Paul Seymour, Professor Dan Spielman, Pro- fessor Dimitrios Thilikos, Martin Demaine, and James Lee for fruitful collaborations and discussions in the areas related to this thesis. I especially thank Paul Seymour for many helpful discussions and for providing a portal into the Graph Minor Theory and revealing some of its hidden structure that we use in this thesis. Thanks go to Professor Daniel Kleitman who served on my committee, read my thesis, and gave me useful comments. Furthermore, I would like to thank the sta® and the faculty of the Department of Mathematics and Computer Science and Arti¯cial Intelligence Laboratory at MIT, especially Kathleen Dickey and Linda Okun, for providing such a nice academic environment. I am also grateful to the researchers at IBM T.J. Wat- son Research Center and Microsoft Research for two great summers in the research industry. My friends here at MIT, namely Reza Alam, Mihai Badoiu, Saeed Bagheri, Mohsen Bahramgiri, Eaman Eftekhary, Nick Harvey, Fardad Hashemi, Susan Ho- henberger, Nicole Immorlica, Ali Khaki¯rooz, Bobby Kleinberg, Mahnaz Maddah, 5 Mahammad Mahdian, Vahab Mirrokni, Eddie Nikolova, Hazhir Rahmandad, Mohsen Razavi, Navid Sabbaghi, Saeed Saremi, Anastasios Sidiropoulos, Ali Tabaei, Mana Taghdiri, David Woodru®, Sergey Yekhanin, and many others played important roles in making my life more enjoyable. Also, I would like to extend my appreciation to thank my previous advisors in University of Waterloo, namely Professor Naomi Nishimura, and in Sharif University of Technology, namely Professor Ebad Mahmood- ian and Professor Mohammad Ghodsi, and my many other friends in Iran, Canada and USA for their warmest friendship. I am blessed to have such excellent companies. Last but not least, my dear parents and my family receive my heartfelt gratitude for their sweetest support and never-ending love. I always feel the warmth of their love, even now that we are so far away. I wish to thank my brother, Mahdi, and my sisters, Monir and Mehri, for being the greatest source of love. This thesis is dedicated to my parents and my family. 6 Contents 1 Introduction and Overview 13 1.1 Graph Terminology . 14 1.2 Graph Classes . 15 1.2.1 De¯nitions of Graph Classes . 15 1.3 Structural Properties . 17 1.3.1 Background . 18 1.3.2 Structure of Single-Crossing-Minor-Free Graphs . 20 1.3.3 Structure of H-Minor-Free Graphs . 20 1.3.4 Structure of Apex-Minor-Free Graphs . 22 1.3.5 Grid Minors . 23 1.4 Bidimensional Parameters/Problems . 24 1.5 Parameter-Treewidth Bounds . 25 1.6 Separator Theorems . 26 1.7 Local Treewidth . 27 1.8 Subexponential Fixed-Parameter Algorithms . 29 1.9 Fixed-Parameter Algorithms for General Graphs . 30 1.10 Polynomial-Time Approximation Schemes . 32 1.11 Half-Integral versus Fractional Multicommodity Flow . 34 1.12 Thesis Structure . 34 2 Approximation Algorithms for Single-Crossing-Minor-Free Graphs 37 2.1 Background . 38 2.1.1 Preliminaries . 38 7 2.1.2 Locally Bounded Treewidth . 39 2.2 Clique-sum Decompositions . 40 2.2.1 Relating Clique Sums to Treewidth and Local Treewidth . 40 2.2.2 Decomposition Algorithm . 41 2.3 Locally Bounded Treewidth of Single-Crossing-Minor-Free Graphs . 47 2.3.1 Bounded Local Treewidth . 47 2.3.2 Local Treewidth and Layer Decompositions . 50 2.4 Approximating Treewidth . 54 2.5 Polynomial-time Approximation Schemes . 58 2.5.1 General Schemes for Approximation on Special Classes of Graphs 58 2.5.2 Approximation Schemes for Single-Crossing-Minor-Free Graphs 59 2.6 Concluding Remarks . 63 3 Exponential Speedup of Fixed-Parameter Algorithms for Single-Crossing- Minor-Free Graphs 65 3.1 Background . 67 3.1.1 Preliminaries . 67 3.2 Fixed-Parameter Algorithms for Dominating Set . 70 3.3 Algorithms for Parameters Bounded by the Dominating-Set Number . 72 3.3.1 Variants of the Dominating Set Problem . 74 3.3.2 Vertex Cover . 75 3.3.3 Edge Dominating Set . 76 3.3.4 Clique-Transversal Set . 77 3.3.5 Maximal Matching .