1. Solving of Graph Coloring Problem with Particle Swarm Optimization Amin Fazel Sharif University of Technology Caro Lucas February 2005 Computer Engineering Department, Sharif University of Technology

2. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 2/20 Introduction Graph Coloring Problem Particle Swarm Optimization Using of PSO for solving GCP Experimental Results Outline

3. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 3/20 Introduction Evolutionary algorithms (EAs) search –Genetic programming (GP), which evolve programs –Evolutionary programming (EP), which focuses on optimizing continuous functions without recombination –Evolutionary strategies (ES), which focuses on optimizing continuous functions with recombination –Genetic algorithms (GAs), which focuses on optimizing general combinatorial problems EAs differ from more traditional optimization techniques –They involve a search from a "population" of solutions, not from a single point

4. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 4/20 Introduction Swarm Intelligence is an AI technique –Is based on social behavior –Applied successfully to solve real-world optimization problems Swarm-like algorithms –Ant Colony Optimization (ACO) –Particle Swarm Optimization (PSO)

5. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 5/20 Introduction PSO shares many similarities with EAs –Population-based –Optimization function –Local and global optima PSO also has dissimilarities to EAs –No evolution operators –Sharing information –PSO is easier to implement

6. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 6/20 Introduction Graph Coloring Problem Particle Swarm Optimization Using of PSO for solving GCP Experimental Results Outline

7. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 7/20 Graph Coloring Problem A proper coloring of a graph G = (V;E) is a function from V to a set C of colors such that any two adjacent vertices have different colors The minimum possible number of colors for which a proper coloring of G exists is called the chromatic number of G. It is NP-complete Has many applications –scheduling and timetabling –telecommunications

8. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 8/20 Introduction Graph Coloring Problem Particle Swarm Optimization Using of PSO for solving GCP Experimental Results Outline

9. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 9/20 Classical PSO PSO applies to concept of social interaction to problem solving A set of moving particles (the swarm) is initially "thrown" inside the search space It was developed in 1995 by James Kennedy and Russ Eberhart It has been applied successfully to a wide variety of search and optimization problems

10. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 10/20 Classical PSO Each particle has the following features: –It has a position and a velocity –It knows its position, and the objective function value for this position –It knows its neighbours, best previous position and objective function value (variant: current position and objective function value) –It remembers its best previous position

11. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 11/20 Classical PSO At each time step –Follow its own way –Go towards its best previous position –Go towards the best neighbour's best previous position, or towards the best neighbour (variant)

12. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 12/20 Classical PSO This compromise is formalized by the following equations: xtxt g best vtvt x t+1

13. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 13/20 Classical PSO The three social/cognitive coefficients respectively quantify: –how much the particle trusts itself now –how much it trusts its experience –how much it trusts its neighbours Social/cognitive coefficients are usually randomly chosen, at each time step

14. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 14/20 Introduction Graph Coloring Problem Particle Swarm Optimization Using of PSO for solving GCP Experimental Results Outline

15. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 15/20 Solving GCP with PSO What we really need for using PSO –a search space of positions/states –a cost/objective function f on S, into a set of values, whose minimums are on the solution states. –an order on C, or, more generally, a semi-order, so that for every pair of elements of C, we can say we have either or

16. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 16/20 Solving GCP with PSO The position of each particle is a sequence of colors –For solving GCP with five vertices –Position vector is N-dimensional vector which N is the number of vertices in the graph V1 V1 V2 V2 V3 V3 V 4 V 5

17. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 17/20 Solving GCP with PSO Position of a particle is Cost function –Conflict is the number of vertices whose colors are the same

18. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 18/20 Introduction Graph Coloring Problem Particle Swarm Optimization Using of PSO for solving GCP Experimental Results Outline

19. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 19/20 Experimental Results Results for random graphs per 5 runs. –Stop conditions: Getting to the chromatic number Or, getting to a maximum iteration number –Population is a very important factor VerticesEdges Chromatic Number Succ (fail) 132210 233310 344310 446410 556210 6101939(1)

20. Thursday, February 19, 2005 Computer Engineering Department Sharif University of Technology 20/20 Introduction Graph Coloring Problem Particle Swarm Optimization Using of PSO for solving GCP Experimental Results Outline

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