Juho Lauri Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds Julkaisu 1428 • Publication 1428 Tampere 2016 Tampereen teknillinen yliopisto. Julkaisu 1428 Tampere University of Technology. Publication 1428 Juho Lauri Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds Thesis for the degree of Doctor of Science in Technology to be presented with due permission for public examination and criticism in Tietotalo Building, Auditorium TB224, at Tampere University of Technology, on the 3rd of November 2016, at 12 noon. Tampereen teknillinen yliopisto - Tampere University of Technology Tampere 2016 ISBN 978-952-15-3836-0 (printed) ISBN 978-952-15-3842-1 (PDF) ISSN 1459-2045 Abstract We study rainbow connectivity of graphs from the algorithmic and graph-theoretic points of view. The study is divided into three parts. First, we study the complexity of deciding whether a given edge-colored graph is rainbow-connected. That is, we seek to verify whether the graph has a path on which no color repeats between each pair of its vertices. We obtain a comprehensive map of the hardness landscape of the problem. While the problem is NP-complete in general, we identify several structural properties that render the problem tractable. At the same time, we strengthen the known NP-completeness results for the problem. We pinpoint various parameters for which the problem is fixed-parameter tractable, including dichotomy results for popular width parameters, such as treewidth and pathwidth. The study extends to variants of the problem that consider vertex-colored graphs and/or rainbow shortest paths. We also consider upper and lower bounds for exact parameterized algorithms. In particular, we show that when parameterized by the k number of colors k, the existence of a rainbow s-t path can be decided in O∗(2 ) time and polynomial space. For the highly related problem of finding a path on which all the k colors appear, i.e., a colorful path, we explain the modest progress over the last twenty years. Namely, we prove that the existence of an algorithm for finding a colorful path in (2 ε)knO(1) time for some ε > 0 disproves the so-called Set Cover Conjecture. − Second, we focus on the problem of finding a rainbow coloring. The minimum number of colors for which a graph G is rainbow-connected is known as its rainbow connection number, denoted by rc(G). Likewise, the minimum number of colors required to establish a rainbow shortest path between each pair of vertices in G is known as its strong rainbow connection number, denoted by src(G). We give new hardness results for computing rc(G) and src(G), including their vertex variants. The hardness results exclude polynomial-time algorithms for restricted graph classes and also fast exact exponential-time algorithms (under reasonable complexity assumptions). For positive results, we show that rainbow coloring is tractable for e.g., graphs of bounded treewidth. In addition, we give positive parameterized results for certain variants and relaxations of the problems in which the goal is to save k colors from a trivial upper bound, or to rainbow connect only a certain number of vertex pairs. Third, we take a more graph-theoretic view on rainbow coloring. We observe upper bounds on the rainbow connection numbers in terms of other well-known graph parameters. Furthermore, despite the interest, there have been few results on the strong rainbow connection number of a graph. We give improved bounds and determine exactly the rainbow and strong rainbow connection numbers for some subclasses of chordal graphs. Finally, we pose open problems and conjectures arising from our work. i Preface Despite the single name on the front cover, this thesis would not have seen the light of day without the help and support of several people. I am sincerely grateful to both my supervisor, professor Tapio Elomaa, and my advisor, D.Sc. Henri Hansen. Tapio routinely handled all the administrative tasks allowing me to concentrate on research. Henri’s lecturing in as early as 2008 inspired me to get curious about theory. We had many fruitful discussions already during the time of my Master’s thesis, and our meetings were no different later on. All the guidance on navigating the academia I received from both Tapio and Henri was invaluable. Unknowingly, the work on this thesis began in January 2013 while I was at Michigan Technological University. At the time, the topic was something that I explored out of curiosity alongside with my studies. Little did I know that the work would evolve to be a foundation for my Master’s thesis, and eventually a part of this thesis as well. I wish to warmly thank professor Melissa Keranen for being an excellent teacher, and for the many discussions and collaboration. I quickly found it stimulating, productive, and fun to work on problems not strictly related to the thesis. The most valuable of such undertakings was a 4-month visit to Aalto University in the Summer of 2014, prior to the beginning of my doctoral studies. I want to thank professor Petteri Kaski for the opportunity to work under his guidance. Our many discussions contributed much to my professional growth. A crucial part of the process was (and is) networking. This might require pushing the boundaries of what you feel comfortable with, but I’m glad that I did. First, I want to thank professor Stefan Szeider for an interesting talk at the 2013 SAT/SMT Summer School, and for a visit to TU Wien in December 2014. Second, I am happy to have met professor Mikko Koivisto in June 2015 at the annual meeting for The Finnish Society for Computer Science in Jyväskylä. I very much enjoyed the later visits, problem solving sessions, and discussions at the University of Helsinki. Third, I thank professor Łukasz Kowalik for his openness and positive attitude. I learned a lot in the whiteboard sessions during my visit to the University of Warsaw in September 2015. I thank professor Yongtang Shi from Nankai University and professor Jukka Suomela from Aalto University for kindly agreeing to pre-examine my thesis. Their comments and careful reading improved this work. Similarly, I thank professor Pinar Heggernes from the University of Bergen for acting as my opponent. Furthermore, I thank all the anonymous reviewers of various journals and conferences for helpful comments and suggestions related to the manuscripts included in this thesis. I thank all my co-authors involved with the projects outside of this thesis as well that I got to meet. In no particular order, many thanks Missy, Petteri, Łukasz, Robert, Eduard, ii iii Henri, and Marzio for sharing your expertise. I also thank Pierre Hauweele for plugging in my code and all the practical help with GraPHedron. The financial support from the Emil Aaltonen Foundation for funding my research between 2015 – 2017 is gratefully acknowledged. I also thank the Finnish Foundation for Technology Promotion (TES) for their support. The much needed balance for work came in the form of friends and family. I have had the pleasure of sharing (two separate) offices with Elina, Henri, and Kalle, all of whom have become more important to me than just co-workers. I also thank other people at the Department of Mathematics for making me feel at home. In particular, I wish to thank Tiina Sävilahti and Kari Suomela for helping me (with a smile) with whatever I happened to need help with. For my many other friends here and around the world: thanks! Most importantly, I am thankful for my parents for all the love and support throughout the years. Mikko, thank you for the countless discussions, and for being a brother I could always look up to. Soffi, thank you for everything. Tampere, October 2016, Juho Lauri To the loving memory of Mari. Contents Abstract i Preface ii List of publications vii 1 Introduction1 1.1 Background................................... 1 1.2 Summary of the main contributions ..................... 5 1.3 Author’s contribution ............................. 5 1.4 The structure of the thesis........................... 6 2 Preliminaries7 2.1 Notation..................................... 7 2.2 Structural graph theory ............................ 7 2.3 Rainbow connections in graphs........................ 10 2.4 Fixed-parameter tractability.......................... 12 2.5 Lower bounds on exact algorithms ...................... 14 3 Hardness of finding rainbow paths 16 3.1 Hardness barriers................................ 16 3.2 A charting of the FPT landscape....................... 19 3.3 On fast algorithms for solving rainbow connectivity............. 21 4 Algorithmic aspects of rainbow coloring graphs 23 4.1 Hardness and lower bounds for rainbow coloring .............. 23 4.2 Graphs with bounded structural parameters................. 28 4.3 Variants of rainbow coloring through parameterization........... 30 5 Bounds on the rainbow connection numbers 32 5.1 Upper bounds via colorings and domination................. 32 5.2 Rainbow coloring block graphs ........................ 33 5.3 Rainbow coloring chordal diametral path graphs .............. 42 6 Conclusions 47 6.1 Conjectures and open problems........................ 47 Appendices 50 v Contents vi Appendix A A compendium of common problems 51 A.1 Rainbow connectivity ............................. 51 A.2 Rainbow coloring................................ 52 A.3 Other problems................................. 53 Bibliography 56 List of publications This thesis consists of the following five publications. They are referred to as [P1] – [P5] in the text. The publications are presented in an order natural for the discussion to follow. For each publication, the author ordering is alphabetical. [P1] Juho Lauri. Further hardness results on rainbow and strong rainbow connectivity. Discrete Applied Mathematics 201 (2016), pp. 191 – 200. [P2] Juho Lauri. Complexity of rainbow vertex connectivity problems for restricted graph classes. Submitted to Discrete Applied Mathematics. [P3] Łukasz Kowalik and Juho Lauri. On finding rainbow and colorful paths. Theoretical Computer Science 628 (2016), pp.