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Approximation Algorithms Under the Worst-Case Analysis and the Smoothed Analysis

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Approximation Algorithms Under the Worst-Case Analysis and the Smoothed Analysis Approximation Algorithms under the Worst-case Analysis and the Smoothed Analysis by Weitian Tong A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Computing Science University of Alberta c Weitian Tong, 2015 Abstract How to evaluate the performance of an algorithm is a very important subject in com- puter science, for understanding its applicability, for understanding the problem which it is applied to, and for the development of new ideas that help to improve the ex- isting algorithms. There are two main factors, i.e. the performance measure and the analysis model, that affect the evaluation of an algorithm. In this thesis, we analyse algorithms with approximation ratio as a measure under the worst-case analysis and the smoothed analysis. In particular, several interesting discrete optimization problems from bioinformatics, networking and graph theory are investigated, and currently the best approximation algorithms are designed for them under the classic analysis model | the worst-case analysis and the more \realistic" model | the smoothed analysis. ii Acknowledgements It would not have been possible to write this thesis without the support and encourage- ment of many people, to whom I wish to express my sincere appreciation. Most of all, I thank my supervisor, Dr. Guohui Lin, for his invaluable guidance and support. It is difficult to summarize how many ways I am grateful to Dr. Lin, who is more like a close friend than a supervisor. Of course, I thank him mostly for his enthusiasm and for always leading my research into the right direction while leaving me a lot of freedom at the same time. I also thank him for his patience, understanding and advice as I balanced my research with other aspects of my life. I would like to express my appreciation to my committee members, Dr. Randy Goebel, Dr. Russell Greiner, Dr. Guohui Lin, Dr. Mohammad R. Salavatipour, Dr. Csaba Szepesv´ariand Dr. Qianping Gu for their time, effort, and precious suggestions that helped my research to a great extend. Additionally, I would like to thank my collabo- rators, Dr. Randy Goebel, Dr. Guohui Lin, Dr. Tian Liu, Dr. Taibo Luo, Dr. Huili Zhang, Dr. Binhai Zhu for their help in my research. I would also like to thank my references Dr. Randy Geobel, Dr. Guohui Lin, Dr. Tian Liu, Dr. Liang Zhao and Dr. Binhai Zhu for supporting me when I was applying for jobs. For personal funding and research grants, I thank the Department of Computing Science at the University of Alberta and Alberta Innovates Technology Futures (AITF) Graduate Studentship. And I also thank Dr. Randy Goebel and Dr. Guohui Lin for their generous funding support from the NSERC Discovery Grants. Moreover, my appreciation goes to my friends and colleagues for their generous support and great companionship, especially the members in the Lin group. Last but not least, I want to express my gratitude and deepest appreciation to my family, especially my parents and my grandparents, for their continuous encouragement and wholehearted support. No matter where I am and how far away from them, they are always there for me. It is my great fortune in life to have them as my source of strength, happiness and love. iii Contents Abstract ii Acknowledgements iii Contents iv List of Figures vii 1 Introduction 1 1.1 Roadmap . 1 1.2 Notations and preliminaries . 3 1.3 Approximation algorithms . 6 1.4 Three analysis methods . 8 1.4.1 Worst-case analysis . 9 1.4.2 Average-case analysis . 10 1.4.3 Smoothed analysis . 10 1.4.3.1 Approximability smoothed analysis . 12 1.4.3.2 Approximability based algorithm design . 13 1.4.3.3 Perturbation models . 14 2 Bandpass Problem1 17 2.1 Introduction . 17 2.2 Preliminary . 19 2.2.1 Algorithm template . 20 2.2.2 Key structure of bandpass . 21 19 BP1 2.3 36 -approximation algorithm ....................... 23 2.3.1 Algorithm description . 23 2.3.2 Performance analysis . 23 227 BP2 2.4 426 -approximation algorithm ...................... 25 2.4.1 Algorithm description . 25 2.4.2 Performance analysis . 26 70 p2 BP3 2.5 128− -approximation algorithm ..................... 34 2.5.1 Algorithm description . 34 2.5.2 Performance analysis . 34 2.6 Conclusions and future work . 42 3 Multiple RNA Interaction (MRIP) Problem2 | An Extension of the Bandpass Problem 44 3.1 Introduction . 44 3.2 MRIP with a known RNA interaction order . 46 1This chapter is based on [53, 98, 99]. 2This chapter is based on [103, 104]. iv Contents 3.3 The general MRIP . 47 3.3.1 NP-hardness . 47 3.3.2 A 0:5-approximation algorithm . 47 3.4 The general MRIP with transitivity . 49 3.4.1 A 0:5328-approximation for disallowing pseudoknots . 49 3.4.2 A 0:5333-approximation for allowing pseudoknots . 53 3.5 Conclusions and future work . 55 4 Exemplar Non-Breakpoint Similarity (ENBS) Problem3 56 4.1 Introduction . 56 4.2 Preliminaries . 58 4.3 Inapproximability result . 59 4.4 An O(n0:5)-approximation algorithm . 61 4.4.1 Algorithm description . 61 4.4.2 Performance analysis . 63 4.5 Conclusions and future work . 64 5 Minimum Common Integer Partition (MCIP) Problem4 66 5.1 Introduction . 66 5.1.1 Known results . 67 5.1.2 Our contributions . 68 5.2 A 6=5-approximation algorithm for 2-MCIP . 68 5.2.1 Preliminaries . 68 5.2.2 Algorithm description . 70 5.2.3 Performance analysis . 71 5.3 Proof of Lemma 5.4 . 74 5.4 A 0:6k-approximation algorithm for k-MCIP . 84 5.5 Conclusions and future work . 84 6 Minimum Independent Dominating Set (MIDS) Problem5 86 6.1 Introduction . 86 6.2 A.a.s. bounds on the independent domination number . 88 6.2.1 An a.a.s. lower bound . 89 6.2.2 An a.a.s. upper bound . 91 6.3 A tail bound on the independent domination number . 94 6.4 Approximating the independent domination number . 95 6.5 Conclusions and future work . 96 7 Trie and Patricia Index Trees6 97 7.1 Introduction . 97 7.2 The smoothed height of Trie . 99 7.2.1 An a.a.s. upper bound . 100 7.2.2 An a.a.s. lower bound . 100 7.3 The smoothed height of Patricia . 104 3This chapter is based on [25]. 4This chapter is based on [105]. 5This chapter is based on [100, 101]. 6This chapter is based on [102]. v Contents 7.3.1 An a.a.s. upper bound . 104 7.3.2 An a.a.s. lower bound . 105 7.4 Conclusions and future work . 106 8 Conclusions and Future Work 107 8.1 Summary . 107 8.2 Future work . 108 Bibliography 109 vi List of Figures 2.1 An illustration of the bandpasses of S2(π∗) (in ovals) and the bandpasses of M1 (in boxes) for grouping purpose. A horizontal line in the figure represents a row, led by its index. Rows that are adjacent in π∗ and/or row pairs of M1 are intentionally ordered adjacently. In this figure, rows ra and r are adjacent in π , denoted as (r ; r ) π , and edge (r ; r ) M b ∗ a b 2 ∗ a b 2 1 as well; the bandpasses between these two rows in S2(π∗) thus belong to B . Edges (r ; r ); (r ; r ); (r ; r ) M , while (r ; r ); (r ; r ) π ; the 1 t i j k ` u 2 1 i j k ` 2 ∗ bandpasses between rows ri and rj and between rows rk and r` in S2(π∗) shown in the figure have their group memberships indicated beside them respectively. 21 2.2 An illustration of moving (vj 1; vj) from Yb to Yb and adding (vj; vj+1) to − 00 Yb, where (1) the dashed lines indicate edges in M1, (2) the thin solid lines indicate edges of that have not been processed, (3) the lines labeled with b (respectively,P b , or b ) indicate edges in Y (respectively, Y , or 0 00 b b0 Y ), and (4) the two curves may contain edges of M . 32 b00 1 2.3 Each color represents a matching and dashed line means the given match- ing. This 4-matching needs to be colored with at least 6 colors. 43 3.1 An illustration of free base, basepair-like structure and pseudoknot-like structure, where the two pairs connected by crossing red dashed lines form to be a pseudoknot-like structure. 45 3.2 An illustration of transitivity property, where the pair connected by green dashed line is the possible interaction induced by the transitivity property. 46 3.3 A high-level description of Approx I. .................... 48 3.4 A high-level description of Approx II..................... 50 3.5 A high-level description of Approx III.................... 53 4.1 An illustration of a simple graph for the reduction. 61 4.2 A high-level description of the approximation algorithm . 62 AENBS 5.1 A high-level description of algorithm Apx65. 71 5.2 The definitions of sub-collections of P3 and Q41 using the set intersecting configurations, where a solid (dashed, respectively)∗ line indicates a firm (possible, respectively) set intersection. 75 5.3 The configuration of the shortest path connecting a; b P P . 76 2 4 [ 5 5.4 The configuration of the shortest path connecting a; b Q31∗ Q41∗ . 80 2 [ 2 5.5 The configuration of the overlapping shortest paths connecting a; b Q41∗ and connecting a; b Q 2............................2 81 0 2 41∗ 7.1 The Trie constructed for s1 = 00001 :::, s2 = 00111 :::, s3 = 01100 :::, s = 01111 :::, s = 11010f:::, s = 11111 ::: .
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