Graph Partitioning

Charles-Edmond Bichot, Institution école Centrale de Lyon.

Patrick Siarry, University Paris-Est Créteil (UPEC).

Introduction xiii
Charles-Edmond Bichot, Patrick Siarry

Chapter 1. General Introduction to Graph Partitioning 1
Charles-Edmond Bichot

1.1. Partitioning 1

1.2. Mathematical notions 2

1.3. Graphs 4

1.4. Formal description of the graph partitioning problem 8

1.5. Objective functions for graph partitioning 11

1.6. Constrained graph partitioning 13

1.7. Unconstrained graph partitioning 14

1.8. Differences between constrained and unconstrained partitioning 16

1.9. From bisection to k-partitioning: the recursive bisection method 17

1.10. NP-hardness of graph partitioning optimization problems 19

1.11. Conclusion 22

1.12. Bibliography 22

Part 1: Graph Partitioning for Numerical Analysis 27

Chapter 2. A Partitioning Requiring Rapidity and Quality: The Multilevel Method and Partitions Refinement Algorithms 29
Charles-Edmond Bichot

2.1. Introduction 29

2.2. Principles of the multilevel method 30

2.3. Graph coarsening 33

2.4. Partitioning of the coarsened graph 37

2.5. Uncoarsening and partitions refinement 40

2.6. The spectral method 52

2.7. Conclusion 59

2.8. Bibliography 60

Chapter 3. Hypergraph Partitioning 65
Cédric Chevalier

3.1. Definitions and metrics 65

3.2. Connections between graphs, hypergraphs, and matrices 67

3.3. Algorithms for hypergraph partitioning 68

3.4. Purpose 72

3.5. Conclusion 77

3.6. Software references 78

3.7. Bibliography 78

Chapter 4. Parallelization of Graph Partitioning 81
François Pellegrini

4.1. Introduction 81

4.2. Distributed data structures 84

4.3. Parallelization of the coarsening phase 87

4.4. Folding 93

4.5. Centralization 95

4.6. Parallelization of the refinement phase 96

4.7. Experimental results 107

4.8. Conclusion 111

4.9. Bibliography 111

Chapter 5. Static Mapping of Process Graphs 115
François Pellegrini

5.1. Introduction 115

5.2. Static mapping models 116

5.3. Exact algorithms 121

5.4. Approximation algorithms 123

5.5. Conclusion 133

5.6. Bibliography 134

Part 2: Optimization Methods for Graph Partitioning 137

Chapter 6. Local Metaheuristics and Graph Partitioning 139
Charles-Edmond Bichot

6.1. General introduction to metaheuristics 140

6.2. Simulated annealing 141

6.3. Iterated local search 149

6.4. Other local search metaheuristics 158

6.5. Conclusion 159

6.6. Bibliography 159

Chapter 7. Population-based Metaheuristics, Fusion-Fission and Graph Partitioning Optimization 163
Charles-Edmond Bichot

7.1. Ant colony algorithms 163

7.2. Evolutionary algorithms 165

7.3. The fusion-fission method 182

7.4. Conclusion 195

7.5. Acknowledgments 196

7.6. Bibliography 196

Chapter 8. Partitioning Mobile Networks into Tariff Zones 201
Mustapha Oughdi, Sid Lamrous, Alexandre Caminada

8.1. Introduction 201

8.2. Spatial division of the network 208

8.3. Experimental results 220

8.4. Conclusion 222

8.5. Bibliography 223

Chapter 9. Air Traffic Control Graph Partitioning Application 225
Charles-Edmond Bichot, Nicolas Durand

9.1. Introduction 225

9.2. The problem of dividing up the airspace 227

9.3. Modeling the problem 231

9.4. Airspace partitioning: towards a new optimization metaheuristic 237

9.5. Division of the central European airspace 240

9.6. Conclusion 246

9.7. Acknowledgments 247

9.8. Bibliography 247

Part 3: Other Approaches to Graph Partitioning 249

Chapter 10. Application of Graph Partitioning to Image Segmentation 251
Amir Nakib, Laurent Najman, Hugues Talbot, Patrick Siarry

10.1. Introduction 251

10.2. The image viewed in graph form 251

10.3. Principle of image segmentation using graphs 254

10.4. Image segmentation via maximum flows 257

10.5. Unification of segmentation methods via graph theory 265

10.6. Conclusions and perspectives 269

10.7. Bibliography 271

Chapter 11. Distances in Graph Partitioning 275
Alain Guénoche

11.1. Introduction 275

11.2. The Dice distance 276

11.3. Pons-Latapy distance 281

11.4. A partitioning method for distance arrays 283

11.5. A simulation protocol 286

11.6. Conclusions 292

11.7. Acknowledgments 293

11.8. Bibliography 293

Chapter 12. Detection of Disjoint or Overlapping Communities in Networks 297
Jean-Baptiste Angelelli, Alain Guénoche, Laurence Reboul

12.1. Introduction 297

12.2. Modularity of partitions and coverings 299

12.3. Partitioning method 301

12.4. Overlapping partitioning methods 307

12.5. Conclusion 311

12.6. Acknowledgments 312

12.7. Bibliography 312

Chapter 13. Multilevel Local Optimization of Modularity 315
Thomas Aynaud, Vincent D. Blondel, Jean-Loup Guillaume and Renaud Lambiotte

13.1. Introduction 315

13.2. Basics of modularity 317

13.3. Modularity optimization 319

13.4. Validation on empirical and artificial graphs 327

13.5. Discussion 333

13.6. Conclusion 341

13.7. Acknowledgments 342

13.8. Bibliography 342

Appendix. The Main Tools and Test Benches for Graph Partitioning 347
Charles-Edmond Bichot

A.1. Tools for constrained graph partitioning optimization 348

A.2. Tools for unconstrained graph partitioning optimization 350

A.3. Graph partitioning test benches 351

A.4. Bibliography 354

Glossary 357

List of Authors 361

Index 365

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