Combinatorial problems based on graph partitioning present many practical applications. Examples based on mission planning or routing problems in logistics perfectly illustrate such applications. Nevertheless, these problems are not based on the same partitioning pattern; generally, patterns like cycles, paths, or trees are distinguished. Moreover, these practical applications are not often limited to theoretical problems like the Hamiltonian path problem or the K-node disjoint paths problems. Instead, they combine the graph partitioning problem with several restrictions related to the topology of nodes and arcs. The diversity of implied constraints concerning real-life applications is a practical limit to the resolution of such problems through approaches that consider the partitioning problem independently from each additional restraint.

Xavier Lorca is Associate Professor of Computer Science at the école des Mines de Nantes in France.

PART 1. CONSTRAINT PROGRAMMING AND FOUNDATIONS OF GRAPH THEORY 1

Introduction to Part 1 3

Chapter 1. Introduction to Constraint Programming 5

1.1. What is a variable? 7

1.2. What is a constraint? 8

1.3. What is a global constraint? 10

1.4. What is a propagation algorithm? 11

1.5. What is a consistency level? 14

1.6. What is a constraint solver? 15

1.7. Constraint solvers at work 17

1.8. Organization structure 21

Chapter 2. Graph Theory and Constraint Programming 23

2.1. Modeling graphs with constraint programming 24

2.2. Graph theory at work in constraint programming 34

2.3. Constraint programming at work in graph theory 37

Chapter 3. Tree Graph Partitioning 39

3.1. In undirected graphs 39

3.2. In directed graphs 42

PART 2. CHARACTERIZATION OF TREE-BASED GRAPH PARTITIONING CONSTRAINTS 47

Chapter 4. Tree Constraints in Undirected Graphs 49

4.1. Decomposition 49

4.2. Definition of constraints 51

4.3. A filtering algorithm for the proper-forest constraint 56

4.4. Filtering algorithm for the resource-forest constraint 70

4.5. Summary of undirected tree constraints 80

Chapter 5. Tree Constraints in Directed Graphs 83

5.1. Decomposition 83

5.2. Definition of constraints 86

5.3. Filtering algorithm for the tree constraint 89

5.4. Filtering algorithm for the proper-tree constraint 96

5.5. Summary of tree constraints in directed and undirected graphs 113

Chapter 6. Additional Constraints Linked to Graph Partitioning 117

6.1. Definition of restrictions 118

6.2. Complexity zoo 123

6.3. Interaction between the number of trees and the number of proper trees 129

6.4. Relation of precedence between the vertices of the graph 130

6.5. Relation of conditional precedence 137

6.6. Relation of incomparability between graph vertices 140

6.7. Interactions between precedence and incomparability constraints 143

6.8. Constraining the interior half-degree of each vertex 148

6.9. Summary 151

Chapter 7. The Case of Disjoint Paths 153

7.1. Minimum number of paths in acyclic directed graphs 156

7.2. Minimum number of paths in any directed graph 161

7.3. A path partitioning constraint 169

7.4. Summary 173

Chapter 8. Implementation of a Tree Constraint 175

8.1. Original implementation 176

8.2. Toward a “portable” implementation 181

8.3. Conclusion 191

PART 3. IMPLEMENTATION: TASK PLANNING 193

Introduction to Part 3 195

Chapter 9. First Model in Constraint Programming 199

9.1. Model for the coherence of displacements in space 199

9.2. Modeling resource consumption 200

9.3. Modeling time windows 201

9.4. Modeling coordination constraints between units 202

9.5. Limitations of the proposed model 203

Chapter 10. Advanced Model in Constraint Programming 205

10.1. Modeling the coherence of displacements in space 206

10.2. Modeling resource consumption 208

10.3. Integration of temporal aspects 208

10.4. Propagating time windows 213

PART 4. CONCLUSION AND FUTURE WORK 225

Chapter 11. Conclusion 227

Chapter 12. Perspectives and Criticisms 231

Bibliography 233

Index 239

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