Western University Scholarship@Western Electronic Thesis and Dissertation Repository 4-5-2019 3:30 PM Local Search Approximation Algorithms for Clustering Problems Nasim Samei The University of Western Ontario Supervisor Professor Roberto Solis-Oba The University of Western Ontario Graduate Program in Computer Science A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Nasim Samei 2019 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Theory and Algorithms Commons Recommended Citation Samei, Nasim, "Local Search Approximation Algorithms for Clustering Problems" (2019). Electronic Thesis and Dissertation Repository. 6138. https://ir.lib.uwo.ca/etd/6138 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract In this research we study the use of local search in the design of approximation algorithms for NP-hard optimization problems. For our study we have selected several important and well known clustering problems: k-uncapacitated facility location problem, minimum mutliway cut problem, and constrained maximum k-cut problem. We show that by careful use of the local optimality property of the solutions produced by local search algorithms it is possible to bound the ratio between solutions produced by local search approximation algorithms and optimum solutions. This ratio is what is known as the locality gap of the algorithms. Thep locality gap of our algorithm for the k-uncapacitated facility location problem is 2 + 3 + for any constant > 0. This matches the approximation ratio of the best known algorithm for the problem, proposed by Zhang but our algorithm is simpler. For the minimum multiway cut problem our algorithm has locality gap 2-2/k, which matches the approximation ratio of the isolation heuristic of Dahlhaus et al; however, our experimental results show that in practice our local search algorithm greatly outperforms the isolation heuristic, and furthermore it has comparable performance as that of the three currently best algorithms for the minimum multiway cut problem (by Calinescu et al, Sharma and Vondrak, and Buchbinder et al). For the constrained maximum k-cut problem on hypergraphs we proposed a local search based approximation algorithm with locality gap 1-1/k for a variety of constraints imposed on the k-cuts. The locality gap of our algorithm matches the approximation ratio of the best known algorithm for the max k-cut problem on graphs designed by Vazirani, but our algorithm is more general. Keywords: Combinatorial optimization, Approximation algorithms, Local search, Facility location problem, Multiway cut problem, Max k-cut problem ii Co-Authorship Statement This thesis consists of three research articles that either have already been published or that are currently being considered for publication in specialized journals. The contri- butions of each one of the authors are listed below. Authors are listed in alphabetical order. Chapter 2 includes an article titled "Analysis of a local search algorithm for the k- facility location problem" published in the journal RAIRO-Theoretical Informatics and Applications. Nasim Samei provided some research ideas, she contributed to the analysis of the algorithms and wrote the first draft of the paper. Roberto Solis-Oba contributed with research ideas leading to the design of the algorithm and analysis of the algorithms; he also edited the manuscript. Chapter 3 includes an article titled "A local search algorithm for the multiway cut problem" submitted to Journal of Computer and System Sciences. Andrew Bloch-Hansen implemented the algorithms and performed the experiments; he also wrote the part of the manuscript describing the experimental results. Nasim Samei contributed with research ideas that let to the design of the algorithms and their analysis. She wrote the first draft of the paper. Roberto Solis-Oba contributed with ideas about design and analysis of the algorithm, he also edited the manuscript. Chapter 4 includes an article titled "A local search algorithm for the constrained max k-cut problem on hypergraphs" that is currently under review in Journal of Applied Mathematics and Computing. Nasim Samei contributed with research ideas that let to the design of the algorithms and their analysis. She wrote the first draft of the paper. Roberto Solis-Oba contributed with ideas about design and analysis of the algorithm, he also edited the manuscript. iii Acknowlegements I would like to express my sincere gratitude to my supervisor, Roberto Solis-Oba, for having strong faith in me. Also, I am very thankful for all his tremendous support in all aspects of my professional development. He is a great mentor and knowledgeable supervisor who provided me with wise advice. This thesis would not be possible without his extraordinary perseverance. I am forever indebted to him. I am grateful to my examining committee, Yuri Boykov, Hamada Ghenniwa, Daniel Lizotte and Marc Moreno Maza for accepting to be my examiners and their constructive feedback. I am privileged to have a supportive and understanding family. My greatest thanks to my parents Nastaran and Rasoul for their fantastic support throughout my Ph.D. program especially their great support and patience on my Ph.D. exam date. Also, I am thankful for their great respect for my choices and providing me an environment that I can learn and grow. iv Contents Abstract List of Figures vi List of Tables viii 1 An Introduction to Combinatorial Optimization Problems 1 1.1 Introduction . 1 1.2 Local Search Algorithms . 5 1.3 The Complexity of Computing Local Optimum Solutions . 10 1.4 Local Search in the Design of Approximation Algorithms . 11 1.5 Local Search for Combinatorial Optimization Problems . 13 1.5.1 The Multiprocessor Scheduling Problem . 13 1.5.2 Computing a Spanning Tree with Many Leaves . 14 1.5.3 The k-Set Packing Problem . 14 1.5.4 The Max k-SAT Problem . 15 1.5.5 The Traveling Salesman Problem . 15 1.5.6 The Quadratic Assignment Problem . 16 1.5.7 The k-Set Cover Problem . 17 1.5.8 The Maximum Constraint Satisfaction Problem . 17 1.5.9 The Stable Marriage Problem . 18 1.5.10 Placement of Meters in Networks . 19 1.6 Our Local Search Algorithms . 19 1.6.1 The k-Facility Location Problem . 19 1.6.2 The Multiway Cut Problem . 20 1.6.3 The Constrained Max k-cut Problem on Hypergraphs . 22 1.7 Organization of the Thesis . 25 Bibliography 26 2 Local Search Algorithm for the k-Facility Location Problem 28 2.1 Introduction . 28 2.1.1 Contributions . 29 2.1.2 Organization of the Paper . 29 2.2 A Local Search Algorithm with Multiple Swaps . 30 2.3 First Bound for the Locality Gap . 32 v 2.3.1 Pairing . 33 2.3.2 Analysing the Swaps . 35 Multi-Swaps for Sets Ai and Bi where jAij= jBi|≤ p . 35 Swaps for Sets Ai and Bi where jAij= jBij> p . 36 Single Swaps for Facilities in Sets Ar and Br . 40 Putting It All Together . 40 2.3.3 Special Case When the Ratio of the Biggest Facility Cost to the Smallest Facility Cost Is Less Than p + 1 . 41 2.4 Tight Example . 42 2.5 Scaling the Costs . 43 2.5.1 Bounding the Facility Cost . 44 2.5.2 Bounding the Service Cost . 45 Bibliography 48 3 A Local Search Algorithm for the Multiway Cut Problem 50 3.1 Introduction . 50 3.2 The Local Search Algorithm . 51 3.3 Finding a minimum cost relabel operation . 53 3.4 Algorithm MULTIWAY CUT for the 3-way Cut Problem . 56 3.4.1 First bound . 58 3.4.2 Second Bound . 61 3.4.3 Computing the Approximation Ratio . 62 3.5 The Multiway Cut Problem . 62 3.5.1 First Bound . 63 3.5.2 Second bound . 65 3.5.3 Computing the Approximation Ratio . 66 3.6 Tight Example . 67 3.7 Variations of the Multiway Cut Problem . 68 3.7.1 Nodes that Need to be in the Same Partition . 68 3.7.2 Nodes that Can Only be in Some Partitions . 70 3.8 Experimental Results . 71 3.8.1 Input Data . 71 3.8.2 Test Cases . 72 3.8.3 Results . 73 Input Networks . 74 Graph Characteristics . 75 Running Time . 78 Initial Labeling and Value of .................... 79 3.8.4 Final Observations . 81 3.8.5 Acknowledgements . 81 Bibliography 82 4 Local Search for the Constrained Max k-Cut Problem 84 vi 4.1 Introduction . 84 4.2 The Local Search Algorithm . 87 4.3 Max k-Cut, Max Multiway Cut, and Max Steiner k-Cut Problems . 88 4.4 Max Capacitated k-Cut, Max k-Cut with Given Sizes of Parts . 90 4.5 Directed Max k-Cut Problem . 93 Bibliography 99 5 Conclusions 101 5.1 Main Contributions . 101 5.2 Challenges of Using Local Search in the Design of Approximation Algorithms102 5.3 Strengths of Local Search Algorithms . 103 Bibliography 104 Curriculum Vitae 104 vii List of Figures 1.1 An instance of the traveling salesman problem. 1 1.2 Example of the single swap operation and the corresponding neighborhood function, where F = ff1; f2; f3g and C = fc1; c2; c3; c4; c5g. Clients are connected with solid lines to the facilities that service them, and a client is connected to a facility with a dashed line if that facility was servicing the client before performing the swap but it does not service the client after the swap. 6 1.3 Example of a local optimal solution and a global optimal solution for an instance of the facility location problem.