Solving of G raph C oloring P roblem with P article S warm O ptimization

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

2. Outline • Introduction • Graph Coloring Problem • Particle Swarm Optimization • Using of PSO for solving GCP • Experimental Results Computer Engineering Department Sharif University of Technology

3. 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 Computer Engineering Department Sharif University of Technology

4. 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) Computer Engineering Department Sharif University of Technology

5. 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 Computer Engineering Department Sharif University of Technology

6. Outline • Introduction • Graph Coloring Problem • Particle Swarm Optimization • Using of PSO for solving GCP • Experimental Results Computer Engineering Department Sharif University of Technology

7. 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 Computer Engineering Department Sharif University of Technology

8. Outline • Introduction • Graph Coloring Problem • Particle Swarm Optimization • Using of PSO for solving GCP • Experimental Results Computer Engineering Department Sharif University of Technology

9. 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 Computer Engineering Department Sharif University of Technology

10. 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 Computer Engineering Department Sharif University of Technology

11. 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) Computer Engineering Department Sharif University of Technology

12. xt gbest xt+1 vt Classical PSO • This compromise is formalized by the following equations: Computer Engineering Department Sharif University of Technology

13. 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 Computer Engineering Department Sharif University of Technology

14. Outline • Introduction • Graph Coloring Problem • Particle Swarm Optimization • Using of PSO for solving GCP • Experimental Results Computer Engineering Department Sharif University of Technology

15. Solving GCP with PSO • What we reallyneed 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 Computer Engineering Department Sharif University of Technology

16. V1 V2 V5 V4 V3 Solving GCP with PSO • The position of each particle is a sequence of colors • For solving GCP with five vertices • <1,2,3,4,1> • Position vector is N-dimensional vector which N is the number of vertices in the graph Computer Engineering Department Sharif University of Technology

17. Solving GCP with PSO • Position of a particle is • Cost function • Conflict is the number of vertices whose colors are the same Computer Engineering Department Sharif University of Technology

18. Outline • Introduction • Graph Coloring Problem • Particle Swarm Optimization • Using of PSO for solving GCP • Experimental Results Computer Engineering Department Sharif University of Technology

19. 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 Computer Engineering Department Sharif University of Technology

20. Outline • Introduction • Graph Coloring Problem • Particle Swarm Optimization • Using of PSO for solving GCP • Experimental Results Computer Engineering Department Sharif University of Technology

21. Thanks for your patience !