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Tree-Based Decompositions of Graphs on Surfaces and Applications to the Traveling Salesman Problem

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Tree-Based Decompositions of Graphs on Surfaces and Applications to the Traveling Salesman Problem TREE-BASED DECOMPOSITIONS OF GRAPHS ON SURFACES AND APPLICATIONS TO THE TRAVELING SALESMAN PROBLEM A Thesis Presented to The Academic Faculty by Torsten Inkmann In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Algorithms, Combinatorics, and Optimization School of Mathematics Georgia Institute of Technology April 2008 TREE-BASED DECOMPOSITIONS OF GRAPHS ON SURFACES AND APPLICATIONS TO THE TRAVELING SALESMAN PROBLEM Approved by: Professor William J. Cook, Advisor Professor Robert G. Parker School of Industrial and Systems School of Industrial and Systems Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Robin Thomas, Advisor Professor Xingxing Yu School of Mathematics School of Mathematics Georgia Institute of Technology Georgia Institute of Technology Professor Zdenˇek Dvoˇr´ak Date Approved: December 4th, 2007 School of Mathematics Georgia Institute of Technology ACKNOWLEDGEMENTS I would like to thank both of my advisors, Professors William Cook and Robin Thomas, for their support and guidance. Apart from stating the obvious, namely that having two leading researchers in their respective fields to work with and learn from has been invaluable for my academic work, I would like to thank them in partic- ular for everything else besides thinking about math problems with me. While it may seem self-evident for some to expect dependability, availability, fairness and most of all patience from an advisor, this is not always the case and I would like to express my gratitude to Bill and Robin for excelling also in these disciplines. On the academic side, I would furthermore like to thank Professors Zdenˇek Dvoˇr´ak, Gary Parker and Xingxing Yu for serving on my thesis committee, and Professors Ilya Hicks and Hisao Tamaki for discussions on my research. The past five years at Georgia Tech have been a truly unique experience, and I would like to thank everyone who has been part of that in one way or another. In particular I would like to thank all my friends that I have met in Atlanta and elsewhere, and all my friends from Germany who managed to stay in my life despite my extended period of absence. Doing research in Mathematics can be challenging at times (both Mathematicians and others will easily recognize this as a slight understatement), and there has been no other single more important factor than my family in helping me overcome these challenges. The unconditional support of my sister and my parents is the backbone of my life and this thesis. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS . iii LIST OF TABLES . vi LIST OF FIGURES . vii SUMMARY . viii I INTRODUCTION . 1 1.1 Tree-based decomposition methods and graphs embedded on surfaces 1 1.1.1 Contributions of this thesis . 2 1.2 The Traveling Salesman Problem . 3 1.2.1 Contributions of this thesis . 5 II PRELIMINARIES . 6 2.1 General graph theory . 6 2.2 Graphs on surfaces . 7 2.3 Tree-based decompositions . 12 2.3.1 Branch-width . 12 2.3.2 Tree-width . 13 2.3.3 Carving-width . 15 2.4 The Traveling Salesman Problem . 17 2.4.1 Comb inequalities . 19 2.4.2 Domino parity inequalities . 20 III BRANCH-WIDTH OF EMBEDDED GRAPHS . 23 3.1 Introduction . 23 3.2 Spots, borders and generalized slopes . 25 3.3 Small sets . 36 3.4 Tangles and slopes . 49 3.5 Restraints and the capturing theorem . 52 3.6 Carvings and branch-decompositions . 59 iv 3.7 Conclusion . 67 IV MINOR-MINIMAL PLANAR GRAPHS OF FIXED BRANCH-WIDTH 71 4.1 Introduction . 71 4.2 Proof of Theorem 4.1.1 . 73 4.2.1 Lower bound on branch-width . 75 4.2.2 Upper bound on branch-width . 77 4.3 Other minor-minimal planar graphs of fixed branch-width . 91 V APPLICATIONS OF TREE-DECOMPOSITIONS TO THE TSP . 94 5.1 Separation of comb inequalities using tree-decompositions . 94 5.1.1 The algorithm . 100 5.2 Separating simple DP inequalities using tree-decompo- sitions . 104 5.2.1 The algorithm . 105 5.2.2 Computational results . 113 5.3 Conclusion . 123 APPENDIX A COMPUTATIONAL RESULTS FOR SOLVING STANDARD PROBLEMS ON GRAPHS OF BOUNDED TREE-WIDTH . 126 REFERENCES . 135 v LIST OF TABLES 1 Results of tests `standard', `simpleDP' and `planarDP' . 118 2 Results of tests `planarDP', `planarDP again' and `simpleDP+planarDP'120 3 Overall performance of tree-decomposition based methods . 130 4 Computational results for solving standard problems . 132 vi LIST OF FIGURES 1 Borders on the torus . 27 2 The natural embedding of the 5 × 5 toroidal grid G5 5 on the torus . 38 × 3 Decomposing the 2 × 3 grid . 39 4 Construction of the structure graph SB . 41 5 An example of a cluster C with its structure graph SC on the torus . 43 6 Capturing points x(f1), x(f2) from x(e) for the k × k grid with k = 8 57 7 An embedding of the Petersen graph in P and its planar double cover, the Dodecahedron . 72 8 Configuration in proof of Lemma 4.2.16 . 83 9 Configuration in proof of Corollary 4.2.17 . 85 10 Minor-minimal graphs of branch-width 6 which are not double covers 92 11 The graph G2k+1 for k = 3 . 93 12 Vertex sets and subgraphs at a node t in a tree-decomposition . 95 13 The graph G1=2 (odd edges are black, even edges are red) . 107 14 The graph G0 (for k = 3) from Example 5.2.1 . 112 vii SUMMARY The tree-width and branch-width of a graph are two well-studied examples of parameters that measure how well a given graph can be decomposed into a tree structure. In this thesis we give several results and applications concerning these concepts, in particular if the graph is embedded on a surface. In the first part of this thesis we develop a geometric description of tangles in graphs embedded on a fixed surface (tangles are the obstructions for low branch- width), generalizing a result of Robertson and Seymour. We use this result to establish a relationship between the branch-width of an embedded graph and the carving-width of an associated graph, generalizing a result for the plane of Seymour and Thomas. We also discuss how these results relate to the polynomial-time algorithm to determine the branch-width of planar graphs of Seymour and Thomas, and explain why their method does not generalize to surfaces other than the sphere. We also prove a result concerning the class C2k of minor-minimal graphs of branch- width 2k in the plane, for an integer k ≥ 2. We show that applying a certain construction to a class of graphs in the projective plane yields a subclass of C2k, but also show that not all members of C2k arise in this way if k ≥ 3. The last part of the thesis is concerned with applications of graphs of bounded tree-width to the Traveling Salesman Problem (TSP). We first show how one can solve the separation problem for comb inequalities (with an arbitrary number of teeth) in linear time if the tree-width is bounded. In the second part, we modify an algorithm of Letchford et al. using tree-decompositions to obtain a practical method for separating a different class of TSP inequalities, called simple DP constraints, and study their effectiveness for solving TSP instances. viii CHAPTER I INTRODUCTION 1.1 Tree-based decomposition methods and graphs embed- ded on surfaces In their seminal series of papers resulting in the proof of the Graph Minor Theorem [64], Robertson and Seymour introduced several new ways of decomposing graphs into a tree-like structure. Two of these techniques are tree-decompositions and branch- decompositions, together with the corresponding parameters tree-width and branch- width, measuring how well a graph can be decomposed (for definitions, see Chapter 2.3). These closely related decompositions not only play a major role throughout the proof of the Graph Minor Theorem, but also have applications elsewhere and consequently became the subject of intense research in their own right. Apart from their relevance in graph minor theory, tree- and branch-decompositions also are interesting from an algorithmic point of view: A large class of NP-hard optimization problems on graphs can be solved in polynomial time (or often linear time) provided the width of the input graph is bounded by a constant (for references of results of this type, see 2.3). Although these algorithms are very appealing from a theoretical point of view, especially since they are usually based on a conceptually simple dynamic programming approach, the large constants typically involved present challenges for the practicability of these methods. In the proof of the Graph Minor Theorem, Robertson and Seymour also make heavy use of the concept of embedding graphs on surfaces. Although tree- and branch- decompositions are defined for arbitrary graphs, it turns out that there are close connections between graphs of bounded tree- or branch-width and graphs embedded 1 on surfaces: For example, Robertson and Seymour show that for any fixed planar graph H, a class of graphs G has bounded tree-width if and only if none of the graphs in G contains H as a minor [60]. In [63], Robertson and Seymour establish a characterization of the representativity (for a definition, see 2.2) of a graph embedded in a surface other than the sphere, in terms of `respectful tangles'. A tangle (for a definition see 2.3.2) is an obstruction to low branch-width, i.e. a graph has a tangle of order k (for some integer k ≥ 2) if and only if its branch-width is at least k.
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