Western University Scholarship@Western Electronic Thesis and Dissertation Repository 4-8-2019 10:30 AM Approximation Algorithms for Problems in Makespan Minimization on Unrelated Parallel Machines Daniel R. Page The University of Western Ontario Supervisor Solis-Oba, Roberto 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 © Daniel R. Page 2019 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Discrete Mathematics and Combinatorics Commons, and the Theory and Algorithms Commons Recommended Citation Page, Daniel R., "Approximation Algorithms for Problems in Makespan Minimization on Unrelated Parallel Machines" (2019). Electronic Thesis and Dissertation Repository. 6109. https://ir.lib.uwo.ca/etd/6109 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 A fundamental problem in scheduling is makespan minimization on unrelated parallel ma- chines (RjjCmax). Let there be a set J of jobs and a set M of parallel machines, where every + job J j 2 J has processing time or length pi; j 2 Q on machine Mi 2 M. The goal in RjjCmax is to schedule the jobs non-preemptively on the machines so as to minimize the length of the schedule, the makespan. A ρ-approximation algorithm produces in polynomial time a feasible solution such that its objective value is within a multiplicative factor ρ of the optimum, where ρ is called its approximation ratio. The best-known approximation algorithms for RjjCmax have approximation ratio 2, but there is no ρ-approximation algorithm with ρ < 3=2 for RjjCmax unless P = NP. A longstanding open problem in approximation algorithms is to reconcile this hardness gap. We take a two-pronged approach to learn more about the hardness gap of RjjCmax: (1) find approximation algorithms for special cases of RjjCmax whose approximation ratios are tight (unless P = NP); (2) identify special cases of RjjCmax that have the same 3=2-hardness bound of RjjCmax, but where the approximation barrier of 2 can be broken. This thesis is divided into four parts. The first two parts investigate a special case of RjjCmax called the graph balancing problem when every job has one of two lengths and the machines may have one of two speeds. First, we present 3=2-approximation algorithms for the graph balancing problem with one speed and two job lengths. In the second part of this thesis we give an approximation algorithm for thep graph balancing problem with two speeds and two job lengths with approximation ratio ( 65 + 7)=8 ≈ 1:88278. In the third part of the thesis we present approximation algorithms and hardness of approximation results for two problems called RjjCmax with simple job-intersection structure and RjjCmax with bounded job assignments. We conclude this thesis by presenting algorithmic and computational complexity results for a generalization of RjjCmax where J is partitioned into sets called bags, and it must be that no two jobs belonging to the same bag are scheduled on the same machine. Keywords: Theoretical Computer Science, Approximation Algorithms, Scheduling The- ory, Makespan Minimization, Unrelated Parallel Machines, Restricted Assignment Problem, Graph Balancing Problem, Job-Intersection Graph, Bounded Job Assignments, Bag Constraints ii Co-Authorship Statement I would like to acknowledge Dr. Roberto Solis-Oba, my supervisor as a co-author of the orig- inal articles that form Chapters 3–6. I must also acknowledge Marten Maack as a co-author for some of the results in Chapter 5: Marten Maack made some suggestions to help simplify the algorithm in Chapter 5.2, provided comments on the paper, and proposed the idea for the reduction given in Chapter 5.1. iii Acknowledgements First, I must thank my supervisor Dr. Roberto Solis-Oba. His guidance, insightful ideas, detailed criticism, and honest feedback were helpful in sharpening my skills as a researcher and as a writer. I must also thank him for giving me the trust to explore my research interests in a fairly independent manner, and patiently reviewing all the drafts I would slide under his office door at odd hours of the night sometimes. Thank you so much Roberto for all your help and for giving me the opportunity to work with you. I would like to thank Marten Maack. Marten had shared my office from late 2015 until mid 2016. He came all the way from Kiel, Germany to work with us as a part of his studies. Thank you again for the conversations and the opportunity to think about problems together. Next, I must thank the people of the Department of Computer Science at the University of Western Ontario. The department has been nothing but friendly to me, and is full of people that I would be happy to sit down and have a chat with. While I can say I have made plenty of wonderful acquantices among my peers, I must explicitly name two people that I cannot thank enough for the time they gave me and would happily call my friends, Dr. James Hughes, and Ethan Jackson. I must especially thank James for the opportunities he has given me as a teaching assistant, and especially for the random conversations we would have. Some day guys I am sure we will actually get actual time to sit down and play games. Coming to Western was the first time I ever travelled far away from anywhere resembling home for me, so I was happy for the time we could spend together when we could. In general, I must thank all the students I have ever taught, past or present; thank you for giving me the opportunity to teach and inspire you. In addition, I want to thank all those that helped me over the years, such as my colleagues in the Department of Computer Science at the University of Manitoba. Thank you Dr. Ben Li for first introducing me to makespan minimization on unrelated parallel machines in 2012. I need to thank my friends, even if they are far and wide but very few. In that I want to acknowledge my friends online, the “Super Friends”; in particular, I want to thank Tor Asbjørn, Marcus Duplechan, and Jasper Valentine for all the laughs and support these years. I would like to thank my family. Since being away from Manitoba, my family has grown with new nieces and nephews. Thank you Mom for being there for me and being strong all these years, and for supporting me in my goals even if the light at the end of the tunnel was unclear. During my program, my family lost one of our beloved dogs, Java (2006–2016). She (Java) was my “research buddy” and one of the last fragments reminding me of my Father, Michael William Page (1956–2010). My Father’s pragmatism was one of the things that inspired me to pursue research in approximation algorithms, the algorithms “equivalent” of the things he would build in his spare time to solve tangible problems for others. Finally I must thank Charen´ Rinckens. She made plenty of sacrifices by coming with me to London, Ontario since I began my journey home in Winnipeg in 2012. Our move to London in late 2014 was our first big journey together on our own. Thank you for all the hours of me asking you to patiently listening to me explain my proofs and ideas to help convince myself, and for offering an extra pair of eyes when I might have needed them sometimes. iv Contents Abstract ii Co-Authorship Statement iii Acknowledgements iv List of Figures viii List of Tables x 1 Introduction 1 1.1 Background . 1 1.1.1 Optimization Problems and Some Complexity Theory . 1 1.1.2 Approximation Algorithms . 2 1.1.3 Parallel Machine Scheduling Problems . 3 1.1.4 A ρ-Relaxed Decision Procedure . 5 1.2 Motivation . 6 1.3 Problems Investigated, Our Results, and Overview . 7 2 Literature Review 13 2.1 Scheduling Identical and Uniform Parallel Machines (PjjCmax and QjjCmax) . 13 2.2 Scheduling Unrelated Parallel Machines . 14 2.2.1 Makespan Minimization on Unrelated Parallel Machines (RjjCmax) . 14 Earlier Research . 14 2-Approximation Algorithm of Lenstra, Shmoys, and Tardos . 14 Further Developments . 17 2.2.2 Restricted Assignment Problem (PjMjjCmax) . 17 The Configuration Linear Program and Better Estimates of the Opti- mum Makespan . 18 Scheduling Problems with Processing Set Restrictions . 19 2.2.3 Graph Balancing Problem (PjMj; jMjj ≤ 2jCmax) . 20 7=4-Approximation Algorithm of Ebenlendr, Krcˇal,´ and Sgall for the Graph Balancing Problem . 22 3=2-Hardness of the Graph Balancing Problem . 25 Graph Orientation Problem (PjMj; jMjj = 2jCmax) . 27 2.3 Other Scheduling Problems . 27 v 2.3.1 Scheduling Parallel Machines with Simple Job-Intersection Structure and Bounded Job Assignments . 27 2.3.2 Scheduling with Machine Types . 28 2.3.3 Scheduling Parallel Machines with Bags . 29 2.3.4 Preemptive Scheduling . 29 2.3.5 Precedence-Constrained Scheduling . 30 3 Graph Balancing Problem with Two Job Lengths 31 3.1 First 3=2-Approximation Algorithm . 31 3.1.1 Step 4 . 32 3.1.2 Step 5.1 . 33 3.1.3 Step 5.2 . 36 3.1.4 Step 5.3 . 38 3.1.5 Approximation Ratio . 40 3.2 A Simpler 3=2-Approximation Algorithm . 40 4 Graph Balancing Problem with Two Speeds and Two Job Lengths 46 4.1 Approximation Algorithm for QjMj; p j 2 f`s; `bgjCmax .