Ant Colony Optimization and Constraint Programming

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1 Ant Colony Optimization and Constraint Programming Christine Solnon Series Editor Narendra Jussien WILEY

2 Table of Contents Foreword Acknowledgements xi xiii Chapter 1. Introduction Overview of the book Constraint programming Ant colony optimization Constraint programming with ant colony optimization. 4 Chapter 2. Computational Complexity Complexity of an algorithm Complexity of a problem The V class ТЪеЛЛР class ЛЛР-complete problems ЛЛР-hard problems Undecidable problems Complexity of optimization problems Where the most difficult instances can be found Phase transition Search landscape Solving ЛГР-hard problems in practice Exploitation of particular cases Approximation algorithms Heuristics and metaheuristics Structuring and filtering the search space 24

2.5. Undecidable problems 14 2.2.6. Complexity of optimization problems 15 2.3. Where the most difficult instances can be found 15 2.3.1. Phase transition 16 2.3.2. Search landscape 19 2.4. Solving ЛГР-hard problems in practice 21 2.

3 vi ACO and CP PART I. CONSTRAINT PROGRAMMING 27 Introduction to Part I 29 Chapter 3. Constraint Satisfaction Problems What is a constraint? Definition of a constraint Arity of a constraint and global constraints What is a constraint satisfaction problem? Complexity of CSPs Optimization problems related to CSPs Maximizing constraint satisfaction Constrained optimization The n-queens problem Description of the problem First CSP model Second CSP model Third CSP model Influence of the model on the solution process The stable marriage problem Description of the problem CSP model Randomly generated binary CSPs The car sequencing problem Description of the problem CSP model Discussion 50 Chapter 4. Exact Approaches Construction of a search tree Constraint propagation Forward checking Maintaining arc consistency Ordering heuristics Heuristics for choosing variables Heuristics for choosing values Randomized restart From satisfaction to optimization problems Discussion 65

4. The n-queens problem 37 3.4.1. Description of the problem 37 3.4.2. First CSP model 38 3.4.3. Second CSP model 39 3.4.4. Third CSP model 40 3.4.5.

4 Table of Contents vii Chapter 5. Perturbative Heuristic Approaches Genetic algorithms Basic principles Using GAs to solve CSPs Local search Basic principles Metaheuristics based on LS Using LS to solve CSPs Particle swarm optimization Basic principles Using PSO to solve CSPs Discussion 80 Chapter 6. Constructive Heuristic Approaches Greedy randomized approaches Basic principles Using greedy randomized algorithms to solve CSPs Estimation of distribution algorithms Basic principles Using EDAs to solve CSPs Ant colony optimization Discussion 91 Chapter 7. Constraint Programming Languages Constraint logic programming Constraint programming libraries Constraint-based local search Discussion 99 PART II. ANT COLONY OPTIMIZATION 101 Introduction to Part II 103 Chapter 8. From Swarm Intelligence to Ant Colony Optimization Complex systems and swarm intelligence Searching for shortest paths by ant colonies Ant system and the traveling salesman problem Ill Pheromone structure Construction of a Hamiltonian cycle by an ant 114

Constructive Heuristic Approaches 85 6.1. Greedy randomized approaches 86 6.1.1. Basic principles 86 6.1.2. Using greedy randomized algorithms to solve CSPs.. 88 6.2. Estimation of distribution algorithms 88 6.

5 viii ACO and CP Pheromone updating step Artificial versus real ants Generic ACO framework Pheromone structure and construction graph Construction of combinations by ants Improving combinations with local search Pheromone updating step Parameters of an ACO algorithm 122 Chapter 9. Intensification versus Diversification ACO mechanisms for intensifying the search ACO mechanisms for diversifying the search Balancing intensification and diversification Measures of diversification/intensification The A-branching factor Resampling ratio Similarity ratio 137 Chapter 10. Beyond Static Combinatorial Problems Multi-objective problems Definition of multi-objective problems Solving multi-objective problems with ACO Dynamic optimization problems Definition of dynamic optimization problems Solving dynamic optimization problems with ACO Optimization problems over continuous domains Definition of continuous optimization problems Solving continuous optimization problems with ACO. 148 Chapter 11. Implementation Issues Data structures Data structures associated with pheromone factors Data structures associated with heuristic factors Data structures associated with ants Selection of a component with respect to probabilities Implementation of a local search procedure Computation of diversification/intensification measures Resampling ratio Similarity ratio 158

Intensification versus Diversification 125 9.1. ACO mechanisms for intensifying the search 125 9.2. ACO mechanisms for diversifying the search 127 9.3.

6 Table of Contents ix PART III. CP WITH ACO 161 Introduction to Part III 163 Chapter 12. Sequencing Cars with ACO Notation A first pheromone structure for identifying good car sequences Pheromone structure Construction of a sequence by an ant Pheromone laying step A second pheromone structure for identifying critical cars Pheromone structure Construction of a sequence by an ant Pheromone updating step Combining the two pheromone structures First pheromone structure Second pheromone structure Construction of a sequence by an ant Comparison of the different ACO algorithms Considered algorithms Test suite Parameter settings Experimental results Comparison of ACO with state-of-the-art approaches Considered approaches IDWalk VFLS Experimental set-up Experimental results Discussion 182 Chapter 13. Subset Selection with ACO Subset selection problems Maximum clique Multidimensional knapsack Maximum Boolean satisfiability Maximum constraint satisfaction Minimum vertex cover Maximum common subgraph 188

3.2. Construction of a sequence by an ant 172 12.3.3. Pheromone updating step 172 12.4. Combining the two pheromone structures 173 12.4.1. First pheromone structure 173 12.4.2. Second pheromone structure 173 12.

7 x ACO and CP Edge-weighted fc-cardinality tree Description of Ant-SSP Construction of a combination by an ant Pheromone laying step Instantiations of Ant-SSP with respect to two pheromone strategies The vertex pheromone strategy The clique pheromone strategy Comparison of the two strategies Instantiation of Ant-SSP to solve CSPs Heuristic factor Local search Experimental results Randomly generated binary instances Results on instances of the 2006 solver competition Discussion 202 Chapter 14. Integration of ACO in a CP Language Framework for integrating ACO within a CP library Pheromone strategy Construction of assignments Pheromone updating step Illustration of ACO-CP on the car sequencing problem CSP model Variable ordering heuristic Pheromone strategies Heuristic factor Experimental results Discussion 214 Chapter 15. Conclusion Towards constraint-based ACO search Towards a reactive ACO search 216 Bibliography 219 Index 231

4.2. Local search 197 13.5. Experimental results 197 13.5.1. Randomly generated binary instances 197 13.5.2. Results on instances of the 2006 solver competition.. 199 13.6. Discussion 202 Chapter 14.

Integration of ACO in a Constraint Programming Language

Integration of ACO in a Constraint Programming Language Madjid Khichane 12, Patrick Albert 1, and Christine Solnon 2 1 ILOG 2 LIRIS, UMR 5205 CNRS / University of Lyon ANTS 08 Motivations Ant Colony Optimization