Final Year Presentation in EWU

EVOLUTIONARY ALGORITHM FOR

GRAPH COLORING PROBLEM

Presented by

Robiul Islam

2009-2-60-004

And

Arup Kumar Pramanik

2009-2-60-008

East West University

Date : 28 April, 2013

1

SUPERVISOR

Professor Dr. Mozammel Huq Azad Khan

Dept. of Computer Science & Engineering

East West University

2

KEY TERM

Evolutionary Algorithm

Binary Encoding

Mutation

Adding new population

Deterministic process

3

GRAPH COLORING PROBLEM

A coloring of simple graph is the assignment of a

color to each vertex of the graph so that no two

adjacent vertices are assigned the same color. The

chromatic number of a graph is the least number of

colors needed for a coloring of this graph.

* Well-Known NP-hard Problem

* Two adjacency nodes does not contain

same colour

* Uses minimum number of colours

* Also known as vertex colouring problem

4

EVOLUTIONARY ALGORITHM

Evolutionary algorithms (EA) are search

algorithm based on the mechanics of natural

selection and natural genetics

In every generation, a new set of artificial

creatures (chromosome) is created using bits and

pieces of the fittest of the old, an occasional new

part is tried for good measure.

5

DETAILS OF MYCIEL3.COL GRAPH

Node = 11

Edge = 20

Initialization Color = Maximum Out degree +1, which is upper bound of chromatic number

Represent binary matrix with Initialization Color*Node

Here number of row = 6

Number of column = 11

Row represent Color

Column represent Node

6

ENCODING TECHNIQUE

0 0 1 0 0 1 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 1 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 1 1 0

Vertices

1

2

3

4

5

6

Colors

1 2 3 4 5 6 7 8 9 10 11

7

ALGORITHM DESCRIPTION : FITNESS

0 0 1 0 0 1 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 1 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 1 1 0

Vertices

1

2

3

4

5

6

Colors

1 2 3 4 5 6 7 8 9 10 11

Invalid Color

Valid Color

Unused Color

8

ALGORITHM DESCRIPTION : FITNESS

(CONTINUE)

0 0 1 0 0 1 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 1 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 1 1 0

Invalid Row : 3

Valid Row : 2

Unused Row :1

Fitness = Invalid Row * (maximum out degree+1) + Valid Row

Here myciel3.col data file

Fitness Value = 3*6+2

= 20

1 2 3 4 5 6 7 8 9 10 11

1

2

3

4

5

6

Two edges share same ages those

colors are not valid

9

ALGORITHM DESCRIPTION : CORRECTION

0 0 1 0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 1 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 1 1 0

1 2 3 4 5 6 7 8 9 10 11

1

2

3

4

5

6

Invalid Row : 2

Valid Row : 4


= 16 10


(CONTINUE )

0 0 1 0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 1 1 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 1 1 0

Invalid Row : 1

Valid Row : 5


= 11

1 2 3 4 5 6 7 8 9 10 11

1

2

3

4

5

6

11


(CONTINUE)

0 0 1 0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 1 1 0 0 0 0 0

0 0 0 1 0 0 0 1 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 1 1 0

Invalid Row : 0

Valid Row : 6


= 6

1 2 3 4 5 6 7 8 9 10 11

1

2

3

4

5

6

12

ALGORITHM DESCRIPTION

Remove duplicate chromosome

Copying total number of population to a

temporary population

13

ALGORITHM DESCRIPTION : MUTATION

0 0 1 0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 1 1 0 0 0 0 0

0 0 0 1 0 0 0 1 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 1 1 0

1 2 3 4 5 6 7 8 9 10 11

1

2

3

4

5

6

0 0 1 0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 1

0 0 0 0 1 1 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1 0

1 2 3 4 5 6 7 8 9 10 11

1

2

3

4

5

6

Basic Chromosome Mutated Chromosome

Randomly select one bit in a row with low probability If it’s zero convert into one If it’s one convert into zero 14

ALGORITHM DESCRIPTION : REPAPERING

POPULATION

0 0 1 0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 1

0 0 0 0 1 1 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1 0

1 2 3 4 5 6 7 8 9 10 11

1

2

3

4

5

6

Mutated Chromosome

0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 1

0 1 0 0 1 1 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11

After Reaper Chromosome

15

ALGORITHM DESCRIPTION

Calculate fitness of mutated population

Corrected mutation population

Remove duplicate chromosome of mutated

population

16

MERGING TEMPORARY POPULATION TO

INITIAL POPULATION ACCORDING TO MINIMUM

FITNESS VALUE AND ADDING

Adding new population of initial population replace

worst population

6 5 6 5 6 5

6 4 4 5 6 5

Fitness Value of

Population

Fitness Value of

Mutated Population

4 5 4 5 5 5Fitness Value after

Merge

17

DETERMINISTIC PROCESS

Before After

Valid Color 6 Valid Color 4

18

EXPERIMENTAL RESULT

Data File (𝐺) Node Edge 𝜒(𝐺) EAGCP [1] [2] [3]

myciel3.col 11 20 4 4 4 4 4

myciel4.col 23 71 5 5 5 5 5

queen5_5.col 25 160 5 5 5 5 5

queen6_6.col 36 290 7 8 7 7 7

myciel5.col 47 236 6 6 6 6 6

huck.col 74 301 11 11 11 11 11

jean.col 80 254 10 10 10 10 10

anna.col 138 493 11 12 11 11 11

david.col 87 406 11 12 11 11 11

19

EXPERIMENTAL RESULT

Data File :myciel3.col

Node: 11

Edge: 20

Generation: 51

Chi (G): 4

EAGCP: 4

Population Size: 50

Maximum Color: 6

Mutation Probability: 10%

Additional Probability: 10%20

EXPERIMENTAL RESULT (CONTINUE)

Date File : myciel4.col

Node: 23

Edge: 71

Generation: 163

Chi (G): 5

EAGCP: 5

Population Size: 50

Maximum Color: 12




Date File : myciel5.col

Node: 47

Edge: 236

Generation: 765

Chi (G): 6

EAGCP: 6

Population Size: 150

Maximum Color: 24




Data File: queen5_5.col

Node: 25

Edge: 160

Generation: 845

Chi (G): 5

EAGCP: 5


Maximum Color: 17




Date File : huck.col

Node: 71

Edge: 301

Generation: 1897

Chi (G): 11

EAGCP: 11


Maximum Color: 54



CONCLUSION

In this thesis work we have focused on a better

minimize chromatic number with proper EA step

for GCP

It helps mutation, evaluate, immune system and

also reduce colour dynamically

Our best result of large dataset is huck.col which

has 71 node and 301 edge and find expected color

in this graph

25

FUTURE WORK

Used our algorithm in different stranded data set

Optimal result of large data set

Reducing the time complexity

26

THANK YOU

27

Final Year Presentation in EWU

Engineering

Transcript of Final Year Presentation in EWU