Springer Optimization and Its Applications VOLUME 62 Managing Editor Panos M. Pardalos (University of Florida) Editor–Combinatorial Optimization Ding-Zhu Du (University of Texas at Dallas) Advisory Board J. Birge (University of Chicago) C.A. Floudas (Princeton University) F. Giannessi (University of Pisa) H.D. Sherali (Virginia Polytechnic and State University) T. Terlaky (McMaster University) Y. Ye (Stanford University) Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in opti- mization is the constantly increasing emphasis on the interdisciplinary na- ture of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics, and other sciences. The series Springer Optimization and Its Applications publishes under- graduate and graduate textbooks, monographs and state-of-the-art exposi- tory work that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow prob- lems, stochastic optimization, optimal control, discrete optimization, multi- objective programming, description of software packages, approximation techniques and heuristic approaches. For further volumes: http://www.springer.com/series/7393
Transcript
Springer Optimization and Its Applications
VOLUME 62
Managing EditorPanos M. Pardalos (University of Florida)
Editor–Combinatorial OptimizationDing-Zhu Du (University of Texas at Dallas)
Advisory BoardJ. Birge (University of Chicago)C.A. Floudas (Princeton University)F. Giannessi (University of Pisa)H.D. Sherali (Virginia Polytechnic and State University)T. Terlaky (McMaster University)Y. Ye (Stanford University)
Aims and ScopeOptimization has been expanding in all directions at an astonishing rateduring the last few decades. New algorithmic and theoretical techniqueshave been developed, the diffusion into other disciplines has proceeded ata rapid pace, and our knowledge of all aspects of the field has grown evenmore profound. At the same time, one of the most striking trends in opti-mization is the constantly increasing emphasis on the interdisciplinary na-ture of the field. Optimization has been a basic tool in all areas of appliedmathematics, engineering, medicine, economics, and other sciences.
The series Springer Optimization and Its Applications publishes under-graduate and graduate textbooks, monographs and state-of-the-art exposi-tory work that focus on algorithms for solving optimization problems andalso study applications involving such problems. Some of the topics coveredinclude nonlinear optimization (convex and nonconvex), network flow prob-lems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of software packages, approximationtechniques and heuristic approaches.
For further volumes:http://www.springer.com/series/7393
Ding-Zhu Du Ker-I Ko Department of Computer Science Department of Computer Science University of Texas at Dallas State University of New York at Stony Brook Richardson, TX 75080 Stony Brook, NY 11794 USA USA [email protected][email protected] Xiaodong Hu Institute of Applied Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China [email protected]
ISSN 1931-6828 ISBN 978-1-4614-1700-2 e-ISBN 978-1-4614-1701-9 DOI 10.1007/978-1-4614-1701-9 Springer New York Dordrecht Heidelberg London
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An approximation algorithm is an efficient algorithm that produces solutions to anoptimization problem that are guaranteed to be within a fixed ratio of the optimalsolution. Instead of spending an exponential amount of time finding the optimalsolution, an approximation algorithm settles for near-optimal solutions within poly-nomial time in the input size. Approximation algorithms have been studied since themid-1960s. Their importance was, however, not fully understood until the discov-ery of the NP-completeness theory. Many well-known optimization problems havebeen proved, under reasonable assumptions in this theory, to be intractable, in thesense that optimal solutions to these problems are not computable within polyno-mial time. As a consequence, near-optimal approximation algorithms are the bestone can expect when trying to solve these problems.
In the past decade, the area of approximation algorithms has experienced an ex-plosive rate of growth. This growth rate is partly due to the development of relatedresearch areas, such as data mining, communication networks, bioinformatics, andcomputational game theory. These newly established research areas generate a largenumber of new, intractable optimization problems, most of which have direct appli-cations to real-world problems, and so efficient approximate solutions to them areactively sought after.
In addition to the external, practical need for efficient approximation algorithms,there is also an intrinsic, theoretical motive behind the research of approximationalgorithms. In the design of an exact-solution algorithm, the main, and often only,measure of the algorithm’s performance is its running time. This fixed measure of-ten limits our choice of techniques in the algorithm’s design. For an approximationalgorithm, however, there is an equally important second measure, that is, the per-formance ratio of the algorithm, which measures how close the approximation al-
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vi Preface
gorithm’s output is to the optimal solution. This measure adds a new dimension tothe design and analysis of approximation algorithms. Namely, we can now study thetradeoff between the running time and the performance ratio of approximation algo-rithms, and apply different design techniques to achieve different tradeoffs betweenthese two measures. In addition, new theoretical issues about the approximation toan optimization problem need to be addressed: What is the performance ratio of anapproximation algorithm for this problem based on certain types of design strategy?What is the best performance ratio of any polynomial-time approximation algorithmfor this problem? Does the problem have a polynomial-time approximation schemeor a fully polynomial-time approximation scheme? These questions are not only ofsignificance in practice for the design of approximation algorithms; they are also ofgreat theoretical interest, with intriguing connections to the NP-completeness the-ory.
Motivated by these theoretical questions and the great number of newly discov-ered optimization problems, people have developed many new design techniquesfor approximation algorithms, including the greedy strategy, the restriction method,the relaxation method, partition, local search, power graphs, and linear and semidef-inite programming. A comprehensive survey of all these methods and results in asingle book is not possible. We instead provide in this book an intensive study of themain methods, with abundant applications following our discussion of each method.Indeed, this book is organized according to design methods instead of applicationproblems. Thus, one can study approximation algorithms of the same nature to-gether, and learn about the design techniques in a more unified way. To this end, thebook is arranged in the following way: First, in Chapter 1, we give a brief introduc-tion to the concept of NP-completeness and approximation algorithms. In Chapter2, we give an in-depth analysis of the greedy strategy, including greedy algorithmswith submodular potential functions and those with nonsubmodular potential func-tions. In Chapters 3, 4, and 5, we cover various restriction methods, including par-tition and Guillotine cut methods, with applications to many geometric problems.In the next four chapters, we study the relaxation methods. In addition to a generaldiscussion of the relaxation method in Chapter 6, we devote three chapters to ap-proximation algorithms based on linear and semidefinite programming, includingthe primal-dual schema and its equivalence with the local ratio method. Finally, inChapter 10, we present various inapproximability results based on recent work inthe NP-completeness theory. A number of examples and exercises are provided foreach design technique. They are drawn from diverse areas of research, includingcommunication network design, optical networks, wireless ad hoc networks, sensornetworks, bioinformatics, social networks, industrial engineering, and informationmanagement systems.
This book has grown out of lecture notes used by the authors at the Universityof Minnesota, University of Texas at Dallas, Tsinghua University, Graduate Schoolof Chinese Academy of Sciences, Xi’an Jiaotong University, Zhejiang University,East China Normal University, Dalian University of Technology, Xinjiang Univer-sity, Nankai University, Lanzhou Jiaotong University, Xidian University, and HarbinInstitute of Technology. In a typical one-semester class for first-year graduate stu-
Preface vii
dents, one may cover the first two chapters, one or two chapters on the restrictionmethod, two or three chapters on the relaxation method, and Chapter 10. With moreadvanced students, one may also teach a seminar course focusing on one of thegreedy, restriction, or relaxation methods, based on the corresponding chapters ofthis book and supplementary material from recent research papers. For instance, aseminar on combinatorial optimization emphasizing approximations based on linearand semidefinite programming can be organized using Chapters 7, 8, and 9.
This book has benefited much from the help of our friends, colleagues, and stu-dents. We are indebted to Peng-Jun Wan, Weili Wu, Xiuzhen Cheng, Jie Wang, Yin-feng Xu, Zhao Zhang, Deying Li, Hejiao Huang, Hong Zhu, Guochuan Zhang, WeiWang, Shugang Gao, Xiaofeng Gao, Feng Zou, Ling Ding, Xianyue Li, My T. Thai,Donghyun Kim, J. K. Willson, and Roozbeh Ebrahimi Soorchaei, who made much-valued suggestions and corrections to the earlier drafts of the book. We are alsograteful to Professors Frances Yao, Richard Karp, Ronald Graham, and Fan Chungfor their encouragement. Special thanks are due to Professor Andrew Yao and theInstitute for Theoretical Computer Science, Tsinghua University, for the generoussupport and stimulating environment they provided for the first two authors duringtheir numerous visits to Tsinghua University.
Dallas, Texas Ding-Zhu DuStony Brook, New York Ker-I KoBeijing, China Xiaodong HuAugust 2011
Contents
Preface v
1 Introduction 11.1 Open Sesame 11.2 Design Techniques for Approximation Algorithms 81.3 Heuristics Versus Approximation 131.4 Notions in Computational Complexity 141.5 NP-Complete Problems 171.6 Performance Ratios 23Exercises 28Historical Notes 33
3 Restriction 813.1 Steiner Trees and Spanning Trees 823.2 k-Restricted Steiner Trees 863.3 Greedy k-Restricted Steiner Trees 89
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3.4 The Power of Minimum Spanning Trees 1023.5 Phylogenetic Tree Alignment 110Exercises 115Historical Notes 121
4 Partition 1234.1 Partition and Shifting 1234.2 Boundary Area 1294.3 Multilayer Partition 1364.4 Double Partition 142
4.4.1 A Weighted Covering Problem 1424.4.2 A 2-Approximation for WDS-UDG on a Small Cell 1464.4.3 A 6-Approximation for WDS-UDG on a Large Cell 1514.4.4 A (6 + ε)-Approximation for WDS-UDG 155
4.5 Tree Partition 157Exercises 160Historical Notes 164
6 Relaxation 2116.1 Directed Hamiltonian Cycles and Superstrings 2116.2 Two-Stage Greedy Approximations 2196.3 Connected Dominating Sets in Unit Disk Graphs 2236.4 Strongly Connected Dominating Sets in Digraphs 2286.5 Multicast Routing in Optical Networks 2356.6 A Remark on Relaxation Versus Restriction 238Exercises 240Historical Notes 243
7 Linear Programming 2457.1 Basic Properties of Linear Programming 2457.2 Simplex Method 2527.3 Combinatorial Rounding 2597.4 Pipage Rounding 2677.5 Iterated Rounding 2727.6 Random Rounding 280
Contents xi
Exercises 289Historical Notes 295
8 Primal-Dual Schema and Local Ratio 2978.1 Duality Theory and Primal-Dual Schema 2978.2 General Cover 3038.3 Network Design 3108.4 Local Ratio 3158.5 More on Equivalence 325Exercises 332Historical Notes 336
