Graph partitioning is a theoretical subject with applications in many areas, principally: numerical analysis, programs mapping onto parallel architectures, image segmentation, VLSI design. During the last 40 years, the literature has strongly increased and big improvements have been made.
This book brings together the knowledge accumulated during many years to extract both theoretical foundations of graph partitioning and its main applications.
Table of Contents:
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
About the Author :
Charles-Edmond Bichot is Associate Professor at École Centrale de Lyon in France. His work always focuses on the graph partitioning optimization problem and its applications.
Patrick Siarry is Professor in automatics and informatics at Paris-Est University in France. His main research interests are the applications of new stochastic global optimization heuristics to various engineering fields.
