Geometric Crossover for Multiway Graph Partitioning

Yong-Hyuk Kim, Yourim Yoon,

Alberto Moraglio, and Byung-Ro Moon

Contents

• Multiway graph partitioning

• Geometric crossover– Hamming distance– Labeling-independent distance

• Fitness landscape analysis

• Experimental results

• Conclusions

Multiway Graph Partitioning Problem

Multiway Graph Partitioning

Cut size : 5

Multiway Graph Partitioning

Cut size : 6

Geometric Crossover

Geometric Crossover

• Line segment

• A binary operator GX is a geometric crossover if all offspring are in a segment between its parents.

• Geometric crossover is dependent on the metric .

x y

Geometric Crossover

• The traditional n-point crossover is geometric under the Hamming distance.

10110

11011

A

B

A

B

11010X

X2

1

3

H(A,X) + H(X,B) = H(A,B)

K-ary encoding and Hamming distance

• Redundant encoding– Hamming distance is not natural.

1

2

4

6

7

35

1 1 2 2 2 3 3

2233311

3311122

2211133

1122233

1133322

6 different representations

Labeling-independent Distance

• Given two K-ary encoding, and ,

,

where is a metric.

• If the metric is the Hamming distance H, LI can be computed efficiently by the Hungarian method.

Labeling-independent Distance

• A = 1213323, B = 1122233

3322211

2233311

3311122

2211133

1122233

1133322

1 2 1 3 3 2 3 5

4

3

5

4

7

LI(A,B) = 3

N-point LI-GX

• Definition (N-point LI-GX)– Normalize the second parent to the first

under the Hamming distance. Do the normal n-point crossover using the first parent and the normalized second parent.

• The n-point LI-GX is geometric under the labeling-independent metric.

Fitness Landscape Analysis

Distance Distributions

Space E(d)

(all-partition, H) 484.364

(local-optimum, H) 484.369

(all-partition, LI) 429.010

(local-optimum, LI) 274.301

Normalized correlogram

Normalized correlogram

Global Convexity

Hamming distance

Correlation coefficient

-0.11

Global Convexity

Labeling-independent distance

Correlation coefficient

0.79

Experimental Results

Genetic Framework

• GA + FM variant

• Population size : 50

• Selection– Roulette-wheel proportional selection

• Replacement– Genitor-style replacement

• Steady-state GA

Test Environment

• Data Set– Johnson’s benchmark data– 4 random graphs (G*.*) and 4 random

geometric graphs (U*.*) with 500 vertices.– Used in a number of other graph-

partitioning studies.

• Tests on 32-way and 128-way partitioning

Experimental Results

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

G500.2.5 G500.05 G500.10 G500.20

5pt H-GX GEFM 5pt LI-GX

32-way partitioning

Experimental Results

32-way partitioning

0

1

2

3

4

5

6

7

8

9

U500.05 U500.10 U500.20 U500.40

5pt H-GX GEFM 5pt LI-GX

Experimental Results

128-way partitioning

0

0.5

1

1.5

2

2.5

3

3.5

G500.2.5 G500.05 G500.10 G500.20

5pt H-GX GEFM 5pt LI-GX

Experimental Results

0

0.2

0.4

0.6

0.8

1

1.2

1.4

U500.05 U500.10 U500.20 U500.40

5pt H-GX GEFM 5pt LI-GX

128-way partitioning

Conclusion

• Methodology– Designed a geometric crossover based on th

e labeling independent distance.– Provided evidence for the fact that the labelin

g-independent distance is more suitable for the multiway graph partitioning problem by the fitness landscape analysis.

• Performance– Performed better than existing genetic algorit

hms.