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New Methods in Parameterized Algorithms and Complexity

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New Methods in Parameterized Algorithms and Complexity New Methods in Parameterized Algorithms and Complexity Daniel Lokshtanov Dissertation for the degree of Philosophiae Doctor (PhD) University of Bergen Norway April 2009 i Acknowledgements First of all, I would like to thank my supervisor Pinar Heggernes for her help and guidance, both with my reasearch and with all other aspects of being a graduate student. This thesis would not have existed without her. Already in advance, I would like to thank my PhD committee, Maria Chudnovsky, Rolf Niedermeier and Sverre Storøy. I also want to thank all my co-authors, Noga Alon, Oren Ben-Zwi, Hans Bod- laender, Michael Dom, Michael Fellows, Henning Fernau, Fedor V. Fomin, Petr A. Golovach, Fabrizio Grandoni, Pinar Heggernes, Danny Hermelin, Jan Kra- tochv´ıl, Elena Losievskaja, Federico Mancini, Rodica Mihai, Neeldhara Misra, Matthias Mnich, Ilan Newman, Charis Papadopoulos, Eelko Penninkx, Daniel Raible, Venkatesh Raman, Frances Rosamond, Saket Saurabh, Somnath Sikdar, Christian Sloper, Stefan Szeider, Jan Arne Telle, Dimitrios Thilikos, Carsten Thomassen and Yngve Villanger. A special thanks goes to Saket Saurabh, with whom I have shared many sleepless nights and cups of coffee preparing this work. Thanks to my family, especially to my mother and father. In more than one way, i am here largely because of you. I would also like to thank my friends for filling my life with things other than discrete mathematics. You mean a lot to me, hugs to all of you and kisses to one in particular. Thanks also to the Algorithms group at the University of Bergen, you are a great lot. An extra thanks goes to my table-tennis opponents, even though this thesis exists despite of your efforts to keep me away from it. Finally, thanks to Pasha for being furry and having whiskers. Bergen, April 2009 Daniel Lokshtanov ii Contents 1 Introduction 1 1.1 OverviewoftheThesis ........................ 2 1.2 NotationandDefinitions . 3 I Overview of the Field 11 2 Algorithmic Techniques 13 2.1 BoundedSearchTrees ........................ 13 2.2 Reduction Rules and Branching Recurrences . 14 2.3 IterativeCompression. 16 2.4 Dynamic Programming and Courcelle’s Theorem . 19 2.5 WellQuasiOrdering ......................... 21 2.6 Win/Wins .............................. 22 2.6.1 Bidimensionality . 23 2.7 ColorCoding ............................. 24 2.8 IntegerLinearProgramming . 26 3 Running Time Lower Bounds 31 3.1 NP-hardness vs. ParaNP-hardness . 31 3.2 HardnessforW[1]........................... 32 3.3 HardnessforW[2]........................... 34 4 Kernelization 37 4.1 ReductionRules............................ 38 4.2 CrownReductions .......................... 39 4.3 LPbasedkernelization . 41 5 Kernelization Lower Bounds 43 5.1 CompositionAlgorithms . 43 5.2 Polynomial Parameter Transformations . 46 iii iv II New Methods 49 6 Algorithmic Method: Chromatic Coding 51 6.1 TournamentFeedbackArcSet . 52 6.1.1 ColorandConquer . .. .. 52 6.1.2 Kernelization ......................... 53 6.1.3 ProbabilityofaGoodColoring . 54 6.1.4 SolvingaColoredInstance . 54 6.2 Universal Coloring Families . 57 7 Explicit Identification in W[1]-Hardness Reductions 61 7.1 Hardness for Treewidth-Parameterizations . ... 61 7.1.1 EquitableColoring . 62 7.1.2 CapacitatedDominatingSet . 67 7.2 Hardness for Cliquewidth-Parameterizations . .... 71 7.2.1 GraphColoring—ChromaticNumber . 71 7.2.2 EdgeDominatingSet. 74 7.2.3 HamiltonianCycle . 79 7.2.4 Conclusion........................... 83 8 Kernelization Techniques 85 8.1 Meta-TheoremsforKernelization . 85 8.1.1 ReductionRules ....................... 87 8.1.2 DecompositionTheorems. 94 8.1.3 Kernels ............................ 98 8.1.4 ImplicationsofOurResults . 101 8.1.5 Practicalconsiderations: . 109 8.2 TuringKernels ............................ 110 8.2.1 The Rooted k-Leaf Out-Branching Problem . 111 8.2.2 Bounding Entry and Exit Points . 115 8.2.3 BoundingtheLengthofaPath . 118 9 Kernelization Lower Bounds 125 9.1 Polynomial Parameter Transformations with Turing Kernels . 125 9.2 Explicit Identification in Composition Algorithms . ..... 128 9.2.1 Steiner Tree, Variants of Vertex Cover, and Bounded Rank SetCover ........................... 129 9.2.2 UniqueCoverage . 134 9.2.3 BoundedRankDisjointSets . 136 9.2.4 DominationandTransversals . 137 9.2.5 Numeric Problem: Small Subset Sum . 140 10 Concluding Remarks and Open Problems 143 Chapter 1 Introduction Parameterized Algorithms and Complexity is a natural way to cope with problems that are considered intractable according to the classical P versus NP dichotomy. In practice, large instances of NP-complete problems are solved exactly every day. The reason for why this is possible is that problem instances that occur naturally often have a hidden structure. The basic idea of Parameterized Algo- rithms and Complexity is to extract and harness the power of this structure. In Parameterized Algorithms and Complexity every problem instance comes with a relevant secondary measurement k, called a parameter. The parameter measures how difficult the instance is, the larger the parameter, the harder the instance. The hope is that at least for small values of the parameter, the problem instances can be solved efficiently. This naturally leads to the definition of fixed param- eter tractability. We say that a problem is fixed parameter tractable (FPT) if problem instances of size n can be solved in f(k)nO(1) time for some function f independent of n. Over the last two decades, Parameterized Algorithms and Complexity has established itself as an important subfield of Theoretical Computer Science. Pa- rameterized Algorithms has its roots in the Graph Minors project of Robertson and Seymour [122], but in fact, such algorithms have been made since the early seventees [54, 108]. In the late eightees several FPT algorithms were discov- ered [61, 59] but despite considerable efforts no fixed parameter tractable algo- rithm was found for problems like Independent Set and Dominating Set. In the early ninetees, Downey and Fellows [51, 50] developed a framework for showing infeasibility of FPT algorithms for parameterized problems under cer- tain complexity theoretic assumptions, and showed that the existence of FPT algorithms for Independent Set and Dominating Set is unlikely. The possibility of showing lower as well as upper bounds sparked interest in the field, and considerable progress has been made in developing methods to show tractability and intractability of parameterized problems. Notable methods that have been developed include Bounded Search Trees, Kernelization, Color Coding 1 2 and Iterative Compression. Over the last few years Kernelization has grown into a subfield of its own, due to a connection to polynomial time preprocessing and the techniques that have been developed over the last year for showing kernelization lower bounds [70, 20]. In this thesis we develop new methods for showing upper and lower bounds, both for Parameterized Algorithms and for Kernelization. 1.1 Overview of the Thesis In Section 1.2 we give the necessary notation and definitions for the thesis. In Part I we give an overview of existing techniques in Parameterized Algorithms and Complexity. This part of the thesis is organized as follows. First we cover techniques for algorithm design, then we review existing medthods for show- ing running time lower bounds up to certain complexity-theoretic assumptions. Then we turn our attention to kernelization or polynomial time preprocessing. In Chapter 4 we describe the most well-known techniques for proving kerneliza- tion results. In Chapter 5 we survey the recent methods developed for giving kernelization lower bounds. In Part II we describe new methods in Parameterized Algorithms and Com- plexity that were developed as part of the author’s PhD research. The part is organized similarly to Part I, that is, we first consider algorithmic upper and lower bounds, before turning our attention to upper and lower bounds for ker- nelization. In Chapter 6 we describe a new color-coding based scheme to give subexponential time algorithms for problems on dense structures. In Chapter 7 we illustrate how explicit identification makes it easier to construct hardness re- ductions, and use this kind of reductions to show a trade-off between expressive power and computational tractability. In Chapter 8 we give two results. Our first result is a meta-theorem for kernelization of graph problems restricted to graphs embeddable into surfaces with constant genus. Our second result is a demonstra- tion that a relaxed notion of kernelization called Turing kernelization indeed is more powerful than kernelization in the traditional sense. Finally, in Chapter 9 we show that explicit identification is useful not only in hardness reductions, but also to show kernelization lower bounds. Several such lower bounds are derived by applying the proposed method. Parts of this thesis have been published as conference articles and some parts are excerpts from articles currently under peer review. Below we give an overview of the articles that were used as a basis for this work. Section 2.3 is based on the manuscript Simpler Parameterized Algorithm • for OCT, coauthored with S. Saurabh and S. Sikdar. Section 2.8 is based on an excerpt from the article Graph Layout problems • 3 Parameterized by Vertex Cover, coauthored with M. Fellows, N. Misra, F. A. Rosamond and S. Saurabh. Proceedings of ISAAC 2008, pages 294-305. Section 4.2 is based on an excerpt from the manuscript Hypergraph Coloring • meets Hypergraph Spanning Tree, coauthored with S. Saurabh.
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