Transcript GCP with PSO

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
Outline
• Introduction
• Graph Coloring Problem
• Particle Swarm Optimization
• Using of PSO for solving GCP
• Experimental Results
Thursday,
February 19, 2005
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Sharif University of Technology
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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
Thursday,
February 19, 2005
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Sharif University of Technology
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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)
Thursday,
February 19, 2005
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Sharif University of Technology
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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
Thursday,
February 19, 2005
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Sharif University of Technology
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Outline
 Introduction
• Graph Coloring Problem
• Particle Swarm Optimization
• Using of PSO for solving GCP
• Experimental Results
Thursday,
February 19, 2005
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Sharif University of Technology
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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
Thursday,
February 19, 2005
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Sharif University of Technology
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Outline
 Introduction
 Graph Coloring Problem
• Particle Swarm Optimization
• Using of PSO for solving GCP
• Experimental Results
Thursday,
February 19, 2005
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Sharif University of Technology
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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
Thursday,
February 19, 2005
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Sharif University of Technology
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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
Thursday,
February 19, 2005
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Sharif University of Technology
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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)
Thursday,
February 19, 2005
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Sharif University of Technology
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Classical PSO
• This compromise is formalized by the
following equations:
vt 1  wt vt  c2  pbest,t  xt   c3 gbest,t  xt 

 xt 1  xt  vt 1
xt
gbest
xt+1
vt
Thursday,
February 19, 2005
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Sharif University of Technology
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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
Thursday,
February 19, 2005
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Sharif University of Technology
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Outline
 Introduction
 Graph Coloring Problem
 Particle Swarm Optimization
• Using of PSO for solving GCP
• Experimental Results
Thursday,
February 19, 2005
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Sharif University of Technology
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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
f
states.
S 
C  c i
– 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
• ci
• or
Thursday,
February 19, 2005
 cj
ci  c j
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Sharif University of Technology
15/20
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>
V1
V2
V5
V4
V3
– Position vector is N-dimensional vector which N is
the number of vertices in the graph
Thursday,
February 19, 2005
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Sharif University of Technology
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Solving GCP with PSO
• Position of a particle is
x  n1 , n2 ,, nN 
• Cost function
 N
(ni )
max
i 1
f x   
N
 conflict  p  max (ni )
i 1

if conflict  0
else
– Conflict is the number of vertices whose
colors are the same
Thursday,
February 19, 2005
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Sharif University of Technology
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Outline
 Introduction
 Graph Coloring Problem
 Particle Swarm Optimization
 Using of PSO for solving GCP
• Experimental Results
Thursday,
February 19, 2005
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Sharif University of Technology
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Experimental Results
• Results for random graphs per 5 runs.
– Stop conditions:
• Getting to the chromatic number
• Or, getting to a maximum iteration number
Vertices
Edges
Chromatic
Number
Succ (fail)
1
3
2
2
10
2
3
3
3
10
3
4
4
3
10
4
4
6
4
10
5
5
6
2
10
6
10
19
3
9(1)
– Population is a very important factor
Thursday,
February 19, 2005
Computer Engineering Department
Sharif University of Technology
19/20
Outline
 Introduction
 Graph Coloring Problem
 Particle Swarm Optimization
 Using of PSO for solving GCP
 Experimental Results
Thursday,
February 19, 2005
Computer Engineering Department
Sharif University of Technology
20/20
Thanks for your patience !