solving of g raph c oloring p roblem with p article s warm o ptimization

Solving of Graph Coloring Problem

withParticle Swarm Optimization

Amin FazelSharif University of Technology

Caro Lucas

February 2005

Computer Engineering Department, Sharif University of Computer Engineering Department, Sharif University of TechnologyTechnology

Thursday, February 19, 2005

Computer Engineering DepartmentComputer Engineering DepartmentSharif University of TechnologySharif University of Technology

2/20

• Introduction

• Graph Coloring Problem

• Particle Swarm Optimization

• Using of PSO for solving GCP

• Experimental Results

Outline



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



4/20

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)



5/20

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



6/20

Introduction

• Graph Coloring Problem




Outline



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



8/20

Introduction





Outline



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



10/20

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



11/20

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)



12/20

Classical PSO

• This compromise is formalized by the following equations:

11

,3,21

ttt

ttbestttbestttt

vxx

xgcxpcvwv

xt gbest

vt

xt+1



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



14/20

Introduction


Particle Swarm Optimization



Outline



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

S f C c i

ji cc ci c j



16/20


• 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

V1

V2

V3 V4

V5



17/20


• Position of a particle is

• Cost function

– Conflict is the number of vertices whose colors are the same

else )(max

0 conflict if )(max

1

1

ii

N

ii

N

npconflict

nxf

Nnnnx ,,, 21



18/20

Introduction



Using of PSO for solving GCP


Outline



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

Vertices Edges Chromatic

NumberSucc (fail)

11 33 22 22 1010

22 33 33 33 1010

33 44 44 33 1010

44 44 66 44 1010

55 55 66 22 1010

66 1010 1919 33 9(1)9(1)



20/20

Introduction



Using of PSO for solving GCP

Experimental Results

Outline

Thanks for your patience !

solving of g raph c oloring p roblem with p article s warm o ptimization

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