a novel approach in csp with ga by juhos istvan, phillip tann, toth attila, tezuka masaru

A novel approach in CSP with GA

by

Juhos Istvan, Phillip Tann,

Toth Attila, Tezuka Masaru

EvoNet 2002 - Szeged

Contents

• Constraint Satisfaction

• Problem: Graph colouring - ”an old friend”

• Representation

• GA model

• Results

• Conclusion


Constraint Satisfaction

Constraint Satisfaction Problem (CSP) : <X,D,C> where

• X : variables { x1, …, xn }

• D : domain { D1, …, Dn }

• C : constraints { (x, y) | x,y X }


• X = { x1, x2, x3, x4, x5 }

• D = { red, blue, green,… }

• C = { (x1, x2), (x2, x3),(x3, x4), (x2, x4), (x4, x5) }

(xi,xk) means:

<xi colour> != <xk colour>

Graph colouring


Representation: Graph Colouring

Each column is a vertex and each row is a colour.

Ex:

x1 is colour A (code : 1)

x2 cannot be colour A (code : 0)

Goal: minimize the nb of colours.

How: merge the rows

x1 x2 x3 x4 x5

A 1 0 x x x

B 0 1 0 0 x

C x 0 1 0 x

D x 0 0 1 0

E x x x 0 1


Merge operator

Merging two rows:

1 and X 10 and X 00 and 0 01 and 1 1X and X X1 and 0 not allowed0 and 1 not allowed

A 1 0 x x x

C x 0 1 0 x

A+C 1 0 1 0 x


• Phenotype : merged matrix = nb of colours

• Genotype : merging order = permutation of the rows (D, B, A, E,

C)

• Fitness function :

number of rows in the merged matrix

GA Framework


GA framework cont.

Variation Operators:

• Mutation : swap two members in the permutation

• Crossover : standard crossover not allowed (doesn’t preserve permutations)


GA framework cont.

Solution: order-based crossover [Syswerda]

Select a crossing point;

Parent (Head, Tail);

Reorder Parent1 Tail according to Parent2.A B C D E

E B C A D

B A C D E

B E C A D


The program

• Novel Genetic algorithm• EASEA and EO aided• Written in C++• Compiled and running on Linux• Uses common input DIMACS format


Experimental Setting- Problems considered

- URL: http://mat.gsia.cmu.edu/COLOR/instances.html- Size of the problems

- GA parameters:- Nb of individuals: 100- Mutation probability: 0.3- Crossover probability: 0.8- Nb of fitness evaluations: - Typically 100% known solution is found

- How many runs- Computational effort- Compared with previous works


Results cont.Name Optima No Diff.

parameter

No xover

No Diff.parameter

With xover

With Diff.parameter

no xover

With Diff.parameter

with xover

VertexEdges

Flat300_20 20 42 42 42 42 30021375

Le450_15b 15 19 19 19 19 4508169

Queen11_11 11 15 14 14 14 1213960

Mychel7 8 8 8 8 8 1912360

Mulsol.i.1 49 49 49 49 49 1973925


Conclusion

What we have done:• an algorithm to graph colouring• a CSP algorithm• the idea seems exciting• the results seem good

What remains to be done:• more intensive tests• investigate the mutation and crossover operation• improve the fitness function

Thanks to EvoNet 2002, special thanks to Michele Sebag and Jano van Hemert


Perspectives• Pheromone-like information about

constrained variables• Most constrained variables should be put

first.• What are the most constrained variables ?• Learn which variables are the last ones • Stored in a global vector:

– shared by population, – updated at each generation, – exploited to guide mutation.

a novel approach in csp with ga by juhos istvan, phillip tann, toth attila, tezuka masaru

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