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Graph Modification Problems Related to Graph Classes

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Graph Modification Problems Related to Graph Classes Graph modification problems related to graph classes Federico Mancini Dissertation for the degree of Philosophiae Doctor (PhD) University of Bergen Norway May 2008 i Acknowledgements The first person I need to thank is my supervisor Pinar Heggernes. Without her guidance, encouragement and scolding from time to time, this work would not exist. Thank you for taking me as your student, teaching me so much and believing in me from the very start. I have never told you how much this meant to me, but I hope this thesis can make up for at least some of it. These three years gave me the opportunity to fulfill many of my dreams, and for this I will always be thankful to you. Another person to whom I owe a lot for his unconditional help, even when he hardly knew me, is Marc Bezem. Your support has been critical in many occasions, including when I had to decide whether to apply for this PhD. Thank you for convincing me to do it, or I would have regretted it forever. I would like to thank also all my co-authors Hans L. Bodlaender, Michael R. Fellows, Fedor V. Fomin, Pinar Heggernes, Jan Kratochvil, Daniel Lokshtanov, Charis Papadopoulos, Frances Rosamond and Jan Arne Telle for the fruitful collaborations and interesting discussions. Solving problems alone is boring :-). And speaking of collaborations, I cannot forget Andreas Brandst¨adt, Vang Ban Le, Christian Hundt, Peter Wagner and Nguyen Ngoc Tuy for making the two months at Rostock University extremely enjoyable for me and my family. If my PhD has been such a great experience, it is due mostly to all my col- leagues at the Algorithms Group. I would like to thank in particular Alexey for all the fun time we spent together fishing, hiking, drinking or talking just about anything. All of this before we both got married of course... Joanna, thanks for never letting things get boring, and Morten, one day we will catch that giant cod, I promise. A special thank to Yngve, who is always there with a reasonable an- swer to practically any question one could think about. Maybe too many stupid questions are also the reason why he moved out of our office, but thanks to this I could get a very thoughtful new office mate. Daniel thanks for always making sure that I do not work too much, and Saket, thank you for keeping Daniel out of the office. Serge, Rodica and Daniel, thanks to you too for being around and help me whenever I needed it. I also want to thank my parents for always supporting my choices and making sure that I have everything I need, despite the distance that separates us. Finally I want to dedicate this work to my lovely wife Hilde and our beautiful son Alessandro. Hilde, thanks for taking care of Alessandro on your own while I was away, giving me the chance to focus on this thesis and finish it in time. Bergen, May 2008 Federico Mancini ii Contents Part I 3 1 Introduction 3 2 Notation and definitions 7 2.1 Graphclasses ............................. 9 2.1.1 Perfect graphs and their subclasses . 13 2.1.2 Othergraphclasses. 19 3 Practical applications of graph modification problems 23 3.1 Minimizing the number of modifications . 24 3.1.1 Chordalgraphs ........................ 26 3.1.2 Intervalgraphs ........................ 27 3.1.3 Clustergraphs......................... 29 3.1.4 Planargraphs......................... 31 3.1.5 Connectedgraphs. 31 3.1.6 Graphs characterized by cycles . 33 3.1.7 Comparabilitygraphs. 34 3.2 Minimizing the maximum clique size . 35 3.2.1 TreewidthandPathwidth . 37 3.2.2 Bandwidth........................... 39 3.3 Modificationwithrestrictions . 39 3.3.1 Sandwichproblem . .. .. 40 3.3.2 Extendingcolorings. 41 3.3.3 Probegraphs ......................... 42 4 Methods to solve graph modification problems 45 4.1 Restrictedinputs ........................... 46 4.2 Minimality............................... 47 4.2.1 Characterizing minimality . 52 4.2.2 Vertexincrementalapproach. 59 4.2.3 Extraction and sandwich problem . 65 4.2.4 Minimality for other modification problems . 67 iii iv 4.3 Parameterizedalgorithms . 69 5 Solving problems on modified graphs 77 5.1 Parameterizedgraphfamilies. 77 5.2 Getting more realistic: Fuzzy Graphs . 82 6 Conclusions 85 Part II 107 7 Paper I 8 Paper II 9 Paper III 10 Paper IV 11 Paper V Part I Chapter 1 Introduction This thesis consists of two parts. In the second part the research papers that constitute the new results of the thesis are presented. In this first part we want to put these results in a broader perspective and provide a better background on the general topic of graph modification problems. When speaking about graph modification problems, we mean problems con- cerned with deleting or adding edges or vertices from and to a graph, so that the resulting graph has some specified properties. As graphs can be used to represent various real world and theoretical structures, it is not difficult to see that these modification problems can model a large number of practical applications in sev- eral different fields. Some examples are: networks reliability; numerical algebra; molecular biology; computer vision; and relational databases. It is thus natural that such problems have been widely studied, but often presented in different terms and contexts, so that similar results have been rediscovered independently several times. This is one of the reasons why we want to give an organized presentation of all these different, but sometimes equivalent, works. In Chapter 2 we give basic notation and definitions about graph theory and graph classes. Chapter 3 consists of a brief survey of complexity results, with some discussion of the main motivations behind graph modification problems. In Chapter 4 we provide a more technical insight of various methods used to attack these problems, with special focus on those related to the papers that constitute the second part of this thesis. One main approach (Chapter 4.2) consists in relaxing the objective function and asking for a minimal, rather than a minimum, set of edges or vertices that when added or deleted to/from a graph gives the desired property on the graph. Another one (Chapter 4.3) consists in designing algorithms that are optimized for instances of the problem that can be solved by applying a small amount of modifications. Finally in Chapter 5 we describe two special kinds of graph families on which modification problems have not been studied extensively yet. We give some introductory discussion on the properties of these families and some basic results. Our aim is to point out their connection 3 4 with graph modification problems, and why we think they are worth investigating from this specific point of view. Five papers that the author produced during his PhD studies and that form the second part of this thesis, are the following. • Paper I [130]: Minimal Split Completion of Graphs. Pinar Heggernes and Federico Mancini. (Accepted for publication in Disc. Appl. Math.) This paper deals with the problem of adding an inclusion minimal set of edges to a graph, in order to obtain a split graph. We give an algorithm that solves this problem in time linear in the size of the input. In addition we show that split graphs are sandwich monotone, and, as a consequence, the Minimal Split Extraction problem can be solved in linear time as well. • Paper II [131]: Minimal Comparability Completions of Arbitrary Graphs. Pinar Heggernes, Federico Mancini, and Charis Papadopoulos. (Disc. Appl. Math. 156(1):705–71, 2008.) Here we show that the Minimal Comparability Completion problem is poly- nomial time solvable. The result is interesting considering that comparabil- ity graphs do not have the sandwich monotone property. This gives, in fact, further indication that completion problems into graph classes without this property, are harder to solve. • Paper III [178]: Characterizing and Computing Minimal Cograph Completions. Daniel Lokshtanov, Federico Mancini and Charis Papadopoulos. (The short version of this paper appeared in Springer LNCS Proceedings of FAW’08.) The aim of this work is to characterize minimal cograph completions in order to design an efficient algorithm for the minimality testing. We accomplish this showing that testing whether a cograph completion is minimal, and computing a minimal cograph completion of an arbitrary input graph, can be done in time linear in the size of the completion. • Paper IV [179]: Minimum fill-in and treewidth of split+kv and +ke graphs. Federico Mancini (The short version of this paper appeared in Springer LNCS Proceedings of ISAAC’07.) A parameterized graph class Π+kv is the class consisting of all graphs that can be obtained from some graph in the class Π by adding at most k 5 vertices to it. We study the parameterized complexity of Minimum Fill-in and Treewidth on Split+kv graphs, when the parameter is k. We get the surprising result that, when k = 1, Minimum Fill-in is NP-complete, while Treewidth can be solved in linear time. • Paper V [20]: Clustering with partial information. Hans L. Bodlaender, Michael R. Fellows, Pinar Heggernes, Federico Mancini, Charis Papadopoulos and Frances Rosamond. (The short version of this paper has appeared in Springer LNCS Proceed- ings of MFCS’08.) In this paper we consider graphs that lack some information and we inves- tigate the parameterized complexity of some clustering problems on them. The lack of information is represented in the form of undetermined edges, that can be freely turned into either edges or non-edges. This adds a new level of difficulty to the problems, but also makes the model more realistic. We propose some natural parameters to cope with the hardness deriving from the lack of information, and show various complexity results depending on which parameters are used.
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