1. Geometric Crossover for Multiway Graph Partitioning Yong-Hyuk Kim, Yourim Yoon, Alberto Moraglio, and Byung-Ro Moon

2. Contents Multiway graph partitioning Geometric crossover –Hamming distance –Labeling-independent distance Fitness landscape analysis Experimental results Conclusions

3. Multiway Graph Partitioning Problem

4. Multiway Graph Partitioning Cut size : 5

5. Multiway Graph Partitioning Cut size : 6

6. Geometric Crossover

7. 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

8. Geometric Crossover The traditional n-point crossover is geometric under the Hamming distance. 10110 11011 A B A B 11010 X X 2 1 3 H(A,X) + H(X,B) = H(A,B)

9. K-ary encoding and Hamming distance Redundant encoding –Hamming distance is not natural. 1 2 4 6 7 3 5 1122233 2233311 3311122 2211133 1122233 1133322 6 different representations

10. 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.

11. Labeling-independent Distance A = 1213323, B = 1122233 12133235 4 3 5 4 7 LI(A,B) = 3

12. 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.

13. Fitness Landscape Analysis

14. Distance Distributions SpaceE(d) (all-partition, H)484.364 (local-optimum, H)484.369 (all-partition, LI)429.010 (local-optimum, LI)274.301

15. Normalized correlogram

17. Global Convexity Hamming distance Correlation coefficient -0.11

18. Global Convexity Labeling-independent distance Correlation coefficient 0.79

19. Experimental Results

20. Genetic Framework GA + FM variant Population size : 50 Selection –Roulette-wheel proportional selection Replacement –Genitor-style replacement Steady-state GA

21. 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

22. Experimental Results 32-way partitioning

23. Experimental Results 32-way partitioning

24. Experimental Results 128-way partitioning

25. Experimental Results 128-way partitioning

26. Conclusion Methodology –Designed a geometric crossover based on the labeling independent distance. –Provided evidence for the fact that the labeling-independent distance is more suitable for the multiway graph partitioning problem by the fitness landscape analysis. Performance –Performed better than existing genetic algorithms.