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Concepts of Combinatorial Optimization, Volume 1

ISBN: 978-1-84821-147-6
368 pages
August 2010, Wiley-ISTE
Concepts of Combinatorial Optimization, Volume 1 (1848211473) cover image
Combinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management.

The three volumes of the Combinatorial Optimization series aims to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization.

Concepts of Combinatorial Optimization, is divided into three parts:

  • On the complexity of combinatorial optimization problems, that presents basics about worst-case and randomized complexity;
  • Classical solution methods, that presents the two most-known methods for solving hard combinatorial optimization problems, that are Branch-and-Bound and Dynamic Programming;
  • Elements from mathematical programming, that presents fundamentals from mathematical programming based methods that are in the heart of Operations Research since the origins of this field.
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Preface xiii
Vangelis Th. PASCHOS

PART I. COMPLEXITY OF COMBINATORIAL OPTIMIZATION PROBLEMS 1

Chapter 1. Basic Concepts in Algorithms and Complexity Theory 3
Vangelis Th. PASCHOS

1.1. Algorithmic complexity 3

1.2. Problem complexity 4

1.3. The classes P, NP and NPO 7

1.4. Karp and Turing reductions 9

1.5. NP-completeness 10

1.6. Two examples of NP-complete problems 13

1.7. A few words on strong and weak NP-completeness 16

1.8. A few other well-known complexity classes 17

1.9. Bibliography 18

Chapter 2. Randomized Complexity 21
Jérémy BARBAY

2.1. Deterministic and probabilistic algorithms 22

2.2. Lower bound technique 28

2.3. Elementary intersection problem 35

2.4. Conclusion 37

2.5 Bibliography 37

PART II. CLASSICAL SOLUTION METHODS 39

Chapter 3. Branch-and-Bound Methods 41
Irène CHARON and Olivier HUDRY

3.1. Introduction 41

3.2. Branch-and-bound method principles 43

3.3. A detailed example: the binary knapsack problem 54

3.4. Conclusion 67

3.5. Bibliography 68

Chapter 4. Dynamic Programming 71
Bruno ESCOFFIER and Olivier SPANJAARD

4.1. Introduction 71

4.2. A first example: crossing the bridge 72

4.3. Formalization 75

4.4. Some other examples 79

4.5. Solution 83

4.6. Solution of the examples 88

4.7. A few extensions 90

4.8. Conclusion 98

4.9. Bibliography 98

PART III. ELEMENTS FROM MATHEMATICAL PROGRAMMING 101

Chapter 5. Mixed Integer Linear Programming Models for Combinatorial Optimization Problems 103
Frédérico DELLA CROCE

5.1. Introduction 103

5.2. General modeling techniques 111

5.3. More advanced MILP models 117

5.4. Conclusions 132

5.5. Bibliography 133

Chapter 6. Simplex Algorithms for Linear Programming 135
Frédérico DELLA CROCE and Andrea GROSSO

6.1. Introduction 135

6.2. Primal and dual programs 135

6.3. The primal simplex method 140

6.4. Bland’s rule 145

6.5. Simplex methods for the dual problem 147

6.6. Using reduced costs and pseudo-costs for integer programming 152

6.7. Bibliography 155

Chapter 7. A Survey of some Linear Programming Methods 157
Pierre TOLLA

7.1. Introduction 157

7.2. Dantzig’s simplex method 158

7.3. Duality 162

7.4. Khachiyan’s algorithm 162

7.5. Interior methods 165

7.6. Conclusion 186

7.7. Bibliography 187

Chapter 8. Quadratic Optimization in 0–1 Variables 189
Alain BILLIONNET

8.1. Introduction 189

8.2. Pseudo-Boolean functions and set functions 190

8.3. Formalization using pseudo-Boolean functions 191

8.4. Quadratic pseudo-Boolean functions (qpBf) 192

8.5. Integer optimum and continuous optimum of qpBfs 194

8.6. Derandomization 195

8.7. Posiforms and quadratic posiforms 196

8.8. Optimizing a qpBf: special cases and polynomial cases 198

8.9. Reductions, relaxations, linearizations, bound calculation and persistence 200

8.10. Local optimum 206

8.11. Exact algorithms and heuristic methods for optimizing qpBfs 208

8.12. Approximation algorithms 211

8.13. Optimizing a quadratic pseudo-Boolean function with linear constraints 213

8.14. Linearization, convexification and Lagrangian relaxation for optimizing a qpBf with linear constraints 220

8.15. ��-Approximation algorithms for optimizing a qpBf with linear constraints 223

8.16. Bibliography 224

Chapter 9. Column Generation in Integer Linear Programming 235
Irène LOISEAU, Alberto CESELLI, Nelson MACULAN and Matteo SALANI

9.1. Introduction 235

9.2. A column generation method for a bounded variable linear programming problem 236

9.3. An inequality to eliminate the generation of a 0–1 column 238

9.4. Formulations for an integer linear program 240

9.5. Solving an integer linear program using column generation 243

9.6. Applications 247

9.7. Bibliography 255

Chapter 10. Polyhedral Approaches 261
Ali Ridha MAHJOUB

10.1. Introduction 261

10.2. Polyhedra, faces and facets 265

10.3. Combinatorial optimization and linear programming 276

10.4. Proof techniques 282

10.5. Integer polyhedra and min–max relations 293

10.6. Cutting-plane method 301

10.7. The maximum cut problem 308

10.8. The survivable network design problem 313

10.9. Conclusion 319

10.10. Bibliography 320

Chapter 11. Constraint Programming 325
Claude LE PAPE

11.1. Introduction 325

11.2. Problem definition 327

11.3. Decision operators 328

11.4. Propagation 330

11.5. Heuristics 333

11.6. Conclusion 336

11.7. Bibliography 336

List of Authors 339

Index 343

Summary of Other Volumes in the Series 347

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Vangelis Th. Paschos is Exceptional Professor of Computer Science and Combinatorial Optimization at the University Paris-Dauphine and chairman of the LAMSADE (Laboratory for the Modeling and the Analysis of Decision Aiding Systems). His research interests include the complexity theory, the theory of the polynomial approximation of NP-hard problems, the probabilistic combinatorial optimization, the on-line computation and the exact solution of NP-hard problems. He is the author of more than a hundred and fifty research papers. He is also member of the editorial board of several international scientific journals.
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