A novel approach in CSP with GA

1
A novel approach in CSP with GA

• by
• Juhos Istvan, Phillip Tann,
• Toth Attila, Tezuka Masaru

2
Contents

• Constraint Satisfaction
• Problem Graph colouring - an old friend
• Representation
• GA model
• Results
• Conclusion

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Constraint Satisfaction

• Constraint Satisfaction Problem (CSP) ltX,D,Cgt
  where
• X variables x1, , xn
• D domain D1, , Dn
• C constraints (x, y) x,y ? X

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Graph colouring

• X x1, x2, x3, x4, x5
• D red, blue, green,
• C (x1, x2), (x2, x3),(x3, x4), (x2, x4),
  (x4, x5)
• (xi,xk) means
• ltxi colourgt ! ltxk colourgt

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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
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Merge operator

• Merging two rows
• 1 and X ? 1
• 0 and X ? 0
• 0 and 0 ? 0
• 1 and 1 ? 1
• X and X ? X
• 1 and 0 ? not allowed
• 0 and 1 ? not allowed

A 1 0 x x x

C x 0 1 0 x


AC 1 0 1 0 x
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GA Framework

• 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

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GA framework cont.

• Variation Operators
• Mutation swap two members in the permutation
• Crossover standard crossover not allowed
  (doesnt preserve permutations)

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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

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The program

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

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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

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Results cont.
Name Optima No Diff. parameter No xover No Diff. parameter With xover With Diff. parameter no xover With Diff. parameter with xover Vertex Edges
Flat300_20 20 42 42 42 42 300 21375
Le450_15b 15 19 19 19 19 450 8169
Queen11_11 11 15 14 14 14 121 3960
Mychel7 8 8 8 8 8 191 2360
Mulsol.i.1 49 49 49 49 49 197 3925
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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

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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.