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

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

K-ary encoding and Hamming distance Redundant encoding –Hamming distance is not natural 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 = , B = 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 SpaceE(d) (all-partition, H) (local-optimum, H) (all-partition, LI) (local-optimum, LI)

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 32-way partitioning

Experimental Results 32-way partitioning

Experimental Results 128-way partitioning

Experimental Results 128-way partitioning

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.