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Quadratic Bottleneck Problems QUADRATIC BOTTLENECK PROBLEMS: ALGORITHMS, COMPLEXITY AND RELATED TOPICS by Ruonan Zhang B.Sc., Lanzhou University, 2003 M.Sc., University of New Brunswick, 2005 a Thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy IN THE DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE c Ruonan Zhang 2011 SIMON FRASER UNIVERSITY Fall 2011 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Ruonan Zhang Degree: Doctor of Philosophy Title of Thesis: Quadratic Bottleneck Problems: Algorithms, Complexity and Related Topics Examining Committee: Dr. Tamon Stephen Assistant Professor, Chair Dr. Abraham Punnen Professor, Senior Supervisor Dr. Zhaosong Lu Assistant Professor, Supervisor Dr. Snezana Mitrovic-minic Adjunct Professor, Internal Examiner Dr. Donglei Du Professor, Faculty of Business Administration, University of New Brunswick, External Examiner Date Approved: October 20, 2011 ii Partial Copyright Licence Abstract In this thesis we study the quadratic bottleneck combinatorial optimization problem (QB- COP) which generalizes the well-known class of bottleneck problems. The solvability of QBCOP is linked to that of the linear combinatorial optimization problems with conflict constraints. Various properties and algorithms for these classes of problems are developed and special cases of spanning trees, knapsack type problems, and assignment variations are explored. These problems are shown to be strongly NP-hard even on very special graphs. We then identify polynomially solvable special cases and also develop heuristic algorithms. Experimental results are reported for all the heuristics we developed. As a by-product our work, we have an approximation algorithm for the maximum edge clique partitioning problem, improving the best known performance ratio for the problem. iii Acknowledgments I express my first and deepest thankfulness to Dr. Abraham Punnen - my senior supervisor. Throughout my studies, he has consistently guided my work with great inspiration, enthu- siasm and patience. Absolutely, I would not be able to accomplish this work without him. To me, Abraham is far beyond a resource of knowledge, a supervisor who provide critics, suggestions, and corrections to my work, but he is the one who has shaped my character, my ethic, and my courage on tackling the incoming difficulties. I am so gratitude for being his student and to learn from him on both the scientific and personal levels is surely a life-long process. My special thanks to paper coauthors - Dr. Santosh Kabadi and Dr. Oncan¨ Temel for their great contributions, it has been a great experience working with and learn from them. Then I would like to thank my supervisory committee member Dr. Zhaosong Lu for his feedback and helpful comments. Also acknowledge helps and encouragements provided by Dr. Tamon Stephen, Dr. Natalia Kouzniak, Dr. Randall Pyke in Department of Mathemat- ics, SFU Surrey. I owe thanks to my colleagues in the SFU Optimization Graduate Student Lab, who have generously shared with me their experience to help my work. I would also like to thank everyone at SFU who helped me in the past six years, specially Department of Mathematics. Last but not least, I would like to thank my family and friends. Without their uncondi- tional love and support, I would never be who I am. iv I dedicate this work to my parents - Du, Li and Zhang, Jiansheng. v Contents Approval ii Abstract iii Acknowledgments iv Dedication v Contents vi List of Tables ix List of Figures x List of Abbreviations xi 1 Combinatorial Optimization Problems 1 1.1 Introduction.................................... 1 1.2 Linear Combinatorial Optimization Problems (LCOP) . .......... 2 1.2.1 LCOP with conflict constraints . 3 1.3 Quadratic Combinatorial Optimization Problems (QCOP) ........... 4 1.3.1 The Unconstrained 0-1 Quadratic Programming Problem (UQP) . 5 1.3.2 The Quadratic Assignment Problem (QAP) . ... 6 1.3.3 The Quadratic Minimum Spanning Tree Problem (QMST) . .... 8 1.3.4 TheQuadraticKnapsackProblem(QKP) . 10 1.3.5 Polynomially solvable cases . 12 1.4 Bottleneck Combinatorial Optimization Problems (BCOP) .......... 13 vi 1.4.1 TherelationoftheBCOPandtheLCOP . 14 1.4.2 Thethresholdalgorithm . 15 1.4.3 Thedualitytheorem ............................ 16 1.4.4 Asymptoticresults . .. .. .. .. .. .. .. .. 16 1.5 Quadratic Bottleneck Combinatorial Optimization Problems (QBCOP) . 17 1.6 Contributions................................... 20 2 The Minimum Spanning Tree Problem with Conflict Constraints 24 2.1 Introduction.................................... 24 2.2 Complexity and polynomially solvable cases . ......... 25 2.2.1 The (k,n)-ladder and a spanning tree counting formula . 26 2.2.2 The MSTC on fan-stars, fans, wheels, ladders and (k,n)-ladders . 29 2.2.3 TheMSTConcactus ........................... 31 2.2.4 The MSTC with special conflict graphs . 33 2.3 Integer programming formulations and lower bounds . .......... 35 2.3.1 Improvingthelowerbound . 38 2.4 Upperboundsandheuristics . 41 2.4.1 A simple construction heuristic . 41 2.4.2 LocalSearch ................................ 41 2.4.3 TabuThresholdingHeuristic . 42 2.4.4 TabuSearchHeuristic . 43 2.4.5 LagrangianHeuristic. 43 2.5 Infeasibilitytests .............................. 43 2.6 Computationalresults . 46 3 The Maximum Edge Clique Partitioning Problem 54 3.1 Introduction.................................... 54 3.2 The 2-phase approximate greedy algorithm . ....... 56 3.3 The maximum edge bi-clique partitioning problem . ......... 60 4 The Quadratic Bottleneck Combinatorial Optimization Problem 64 4.1 Introduction.................................... 64 4.2 Weak duality and the quadratic threshold algorithm . .......... 66 4.3 Polynomially solvable cases . ..... 70 vii 4.4 Asymptoticresults ............................... 81 5 The Quadratic Bottleneck Spanning Tree Problem 83 5.1 Introduction.................................... 83 5.2 Computationalcomplexity. 84 5.3 Polynomially solvable special cases and an exact algorithm .......... 85 5.4 Heuristicalgorithms . .. .. .. .. .. .. .. .. 87 5.5 Computationalresults . 90 5.6 Asymptoticresults ............................... 95 6 The Quadratic Bottleneck Knapsack Problems 97 6.1 Introduction.................................... 97 6.2 Computationalcomplexity. 98 6.3 Polynomially solvable special cases and an exact algorithm .......... 99 6.4 Heuristicalgorithms . .. .. .. .. .. .. .. .. 102 6.4.1 Semi-greedyAlgorithm . 103 6.4.2 Improvingasolution . .. .. .. .. .. .. .. .. 103 6.5 Computationalresults . 105 6.6 VariationsoftheQBKP ............................. 109 7 The Quadratic Bottleneck Assignment Problem 113 7.1 Introduction.................................... 113 7.2 Polynomially solvable cases and an exact algorithm . ...........114 7.3 Assignment problem with conflict constraints . .........116 7.3.1 Computational complexity of the APC . 117 7.3.2 HeuristicsoftheAPC . .. .. .. .. .. .. .. ..120 7.3.3 Computational results with the APC algorithms . 124 7.4 HeuristicsfortheQBAP.. .. .. .. .. .. .. .. 127 7.5 ComputationalresultsfortheQBAP . 130 8 Conclusion 134 Bibliography 137 Vita 154 viii List of Tables 2.1 Type1problems-Infeasible. 49 2.2 Type1problems-Feasible . 51 2.3 Type1problems-Feasibilityunknown. ...... 52 2.4 Type2problems .................................. 53 5.1 QBST:generalproblems(partI) . 93 5.2 QBST:generalproblems(partII). ..... 94 5.3 QBST: relative bounds and obj of general problems . ........ 96 5.4 QBST:specialproblems .. .. .. .. .. .. .. .. 96 6.1 Experimentalresults:I. 108 6.2 Experimental results: II (numbers in bold represent the cases where the heuristics produced an optimal solution) . 109 7.1 APC:lowerandupperbounds . 126 7.2 QBAP:lowerandupperbounds . 131 7.3 QBAP-KB:lowerandupperbounds . 132 7.4 BM:lowerandupperbounds . 132 ix List of Figures 2.1 Examples: fan,wheelandfan-star . ..... 25 2.2 Example:ladder .................................. 26 2.3 Examples: (k,n)-ladder .............................. 27 2.4 A fan-star constructed from a 3-SAT problem . ....... 29 2.5 A ladder constructed from a 3-SAT problem . ...... 30 2.6 Example of a graph satisfying conditions of Theorem 15. Here, k = 2 and nodes u and v arethenodestobedeleted.. 35 2.7 Example: Matching heuristic vs Approximate greedy . .......... 40 3.1 Example: edge clique partitioning . ....... 54 3.2 Example: edge bi-clique partitioning . ........ 60 3.3 Example: Greedy algorithm for the Max-EBCP . ...... 63 4.1 Example: dualitygapfortheQBCOP . 67 5.1 QBST: comparison of lower bounds (the number e and f are edge labels) . 89 7.1 G: a collection of 4-cycles constructed from G˜ ..................117 7.2 A graph G and an associated conflict graph satisfying properties (1) and (2) . 119 7.3 TheimprovementsoftheFGC . 127 7.4 The improvements of the TABU-GM . 127 x List of Abbreviations Abbreviation Meaning Page COP combinatorial optimization problem 1 MST minimum spanning tree problem 1 TSP traveling salesman problem 1 AP assignment problem 2 QAP quadratic assignment problem 2 LCOP linear combinatorial optimization problem 2 KP knapsack problem 2 LCOPC LCOP with conflict constraints 3 QCOP quadratic combinatorial optimization problem 4 UQP unconstrained
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