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Abstract Directed Graphs: Fixed-Parameter ABSTRACT Title of dissertation: DIRECTED GRAPHS: FIXED-PARAMETER TRACTABILITY & BEYOND Rajesh Chitnis, Doctor of Philosophy, 2014 Dissertation directed by: Professor MohammadTaghi Hajiaghayi Department of Computer Science Most interesting optimization problems on graphs are NP-hard, implying that (un- less P = NP) there is no polynomial time algorithm that solves all the instances of an NP-hard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The paradigm of parameterized complexity was introduced by Downey and Fellows to allow for a more refined multivariate analy- sis of the running time. In parameterized complexity, each problem comes along with a secondary measure k which is called the parameter. The goal of parameterized com- plexity is to design efficient algorithms for NP-hard problems when the parameter k is small, even if the input size is large. Formally, we say that a parameterized problem is fixed-parameter tractable (FPT) if instances of size n and parameter k can be solved in f (k) nO(1) time, where f is a computable function which does not depend on n. A pa- · rameterized problem belongs to the class XP if instances of size n and parameter k can be solved in f (k) nO(g(k)) time, where f and g are both computable functions. · In this thesis we focus on the parameterized complexity of transversal and connec- tivity problems on directed graphs. This research direction has been hitherto relatively unexplored: usually the directed version of the problems require significantly different and more involved ideas than the ones for the undirected version. Furthermore, for di- rected graphs there are no known algorithmic meta-techniques: for example, there is no known algorithmic analogue of the Graph Minor Theory of Robertson and Seymour for directed graphs. As a result, the fixed-parameter tractability status of the directed ver- sions of several fundamental problems such as MULTIWAY CUT,MULTICUT,SUBSET FEEDBACK VERTEX SET,ODD CYCLE TRANSVERSAL, etc. was open. In the first part of the thesis, we develop the framework of shadowless solutions for a general class of transversal problems in directed graphs. For this class of problems, we reduce the problem of finding a solution in FPT time to that of finding a shadowless solution. Since shadowless solutions have a good (problem-specific) structure, this pro- vides an important first step in the design of FPT algorithms for problems on directed graphs. By understanding the structure of shadowless solutions, we are able to design the first FPT algorithms for the DIRECTED MULTIWAY CUT problem and the SUBSET DIRECTED FEEDBACK VERTEX SET problem. In the second part of the thesis, we present tight bounds on the parameterized com- plexity of well-studied directed connectivity problems such as STRONGLY CONNECTED STEINER SUBGRAPH and DIRECTED STEINER FOREST when parameterized by the number of terminals/terminal pairs. We design new optimal XP algorithms for the afore- mentioned problems, and also prove matching lower bounds for existing XP algorithms. Most of our hardness results hold even if the underlying undirected graph is planar. Finally, we conclude with some open problems regarding the parameterized com- plexity of transversal and connectivity problems on directed graphs. DIRECTED GRAPHS: FIXED-PARAMETER TRACTABILITY & BEYOND by Rajesh Chitnis Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2014 Advisory Committee: Professor MohammadTaghi Hajiaghayi, Chair/Advisor Professor Aravind Srinivasan Professor Dana Nau Professor Subramanian Raghavan, Robert H. Smith School of Business and Institute for Systems Research Professor Erik Demaine, CSAIL, MIT c Copyright by Rajesh Chitnis 2014 Dedication To my sister Ishwari and my parents Sandhya and Hemant for all their love and support ii Acknowledgments First of all, I would like to thank my thesis advisor Prof. MohammadTaghi Haji- aghayi for his excellent scientific guidance over the last 4 years. He has been a wonderful source of problems and ideas. Looking back, I can see his imprint on every aspect of my scientific development. Mohammad always encouraged me to work on problems that I found interesting, even when sometimes they might not have been aligned with his re- search interests. He was always willing to fund me for conferences (even when I did not have a paper to present) and also for research visits - in Summer ’12 he gave me an RA while I was away from Maryland the whole summer on research visits to Hungary and Norway! Thanks Mohammad for everything, and I look forward to our continued collaboration. In grad school I was extremely fortunate to have had a chance to work with various researchers who were very generous with their time and ideas. Daniel Lokshtanov hosted me on a visit to San Diego which included a wonderful hike. I spent a month at Bergen, Norway visiting Fedor Fomin and Petr Golovach which led to our collaboration on the Anchored k-Core problem. I learned a lot about cut problems from Marek Cygan during the six months that he visited us at Maryland. I owe whatever little knowledge about streaming algorithms that I have to Morteza Monemizadeh. Two people have had a sig- nificant impact on me during my PhD studies - Daniel Marx and Saket Saurabh. Daniel has been a wonderful collaborator - I have learnt a lot from his intuition about important separators. Thanks also for hosting me for two months in Budapest. Saket has been a good friend and also the main reason why I began to work in FPT in the first place - in the last semester of my undergrad studies, I attended two courses by Saket on Parameterized Complexity and Advanced Graph Theory. These two courses and spending the ensuing summer working with Saket was enough for me to decide to pursue FPT in my PhD. I also spent a wonderful three months at TTI-Chicago working with Julia Chuzhoy. At Maryland one of the best things that happened to me was meeting people from various countries and getting to know their perspectives on life. I hope it has made me less narrow-minded than I was before. In the department, I would like to thank Samir Khuller and Aravind Srinivasan for their continued encouragement and sound advice throughout my PhD. Many thanks to Aravind Srinivasan, Erik Demaine, Dana Nau, Subramanian Raghavan for serving on my dissertation committee and informative comments on the thesis. I was able to remain oblivious to the intricacies of department paperwork thanks to Jenny Story, Fatima Bangura and Sharron McElroy. Many thanks to all my friends in the department - Ioana, Theo, Ben, Vahid, Reza, Kotaro, Aishwarya, Vikas, Abdul, Jayanta, Govind, Udayan, Kanthi, Manish, Amit, David, Hamid, Faezeh and some others whom I am surely forgetting right now. Doing a PhD becomes very monotonous if one does not have fun people around - many thanks to Varun, Raka, Prat Di, Puneet, Arvind and Vivek for various trips, tasty food and cold beer. Finally, thanks to my Persian group of friends for allowing me into their soccer team and also for the several intramural championships that we won. I was very lucky to pursue my undergraduate studies at a unique institution like Chennai Mathematical Institute (CMI) which provided a lot of academic freedom. After iii misusing it for a while, I later was able to take advantage of it and attend various grad courses to see which subareas interest me more. Prof. K. Narayan Kumar gave me a second chance after I did not perform well in the first course with him, and I am glad for that. My first research interaction was with Prof. Samir Datta. Unfortunately our work together did not lead to a paper, but when my interests became a bit more focused Samir referred me to Prof. Sunil Chandran at IISc, Bangalore. Working with Sunil was a very good experience which resulted in my first publication. On the non-academic side, I want to thank all my friends at CMI (most of whom are part of the bug07 mailing list). Leaving the most important for the last, I want to thanks my parents, sister and my extended family. My extended family has always been very encouraging in all my endeavours, and provide important support to my family in my absence. Talking to friends from high school is always fun and very relaxing. Special mention goes to Harshad, Saurabh, Varun, Ambarish and Neeraj. My parents Sandhya and Hemant have been my role-models in my entire life. Everyday I understand more and more how truly awesome they are as parents. Talking to my sister Ishwari always gives me a different perspective of life. She is very enthusiastic about life and is always able to cheer me up. This thesis is a small token of my love and appreciation for my family - no words or actions can ever come close to thank you for all that you do for me. I gratefully acknowledge the support of the following grants during my PhD stud- ies: NSF CAREER award 1053605, NSF grant CCF-1161626, DARPA/AFOSR grant FA9550-12-1-0423, ONR YIP award N000141110662, DARPA/AFRL award FA8650- 11-1-7162, NSF CAREER award 0844872, European Research Council (ERC) grant 280152 “PARAMTIGHT”, European Research Council (ERC) grant 267959 “Rigorous Theory of Preprocessing”, a Summer International Research Fellowship from University of Maryland, and a Simons Award for Graduate Students in Theoretical Computer Sci- ence. iv Table of Contents List of Figures vii 1 Introduction and Overview1 1.1 Notation...................................1 1.2 Coping with NP-Completeness.......................4 1.2.1 Approximation Algorithms.....................7 1.2.2 Randomized Algorithms......................9 1.2.3 Exact Exponential Algorithms..................
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