Geometric Crossover for Multiway Graph Partitioning

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Geometric Crossover for Multiway Graph
Partitioning

• Yong-Hyuk Kim, Yourim Yoon,
• Alberto Moraglio, and Byung-Ro Moon

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Contents

• Multiway graph partitioning
• Geometric crossover
• Hamming distance
• Labeling-independent distance
• Fitness landscape analysis
• Experimental results
• Conclusions

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Multiway Graph Partitioning Problem
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Multiway Graph Partitioning
Cut size 5
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Multiway Graph Partitioning
Cut size 6
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Geometric Crossover
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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 .

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Geometric Crossover

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

H(A,X) H(X,B) H(A,B)
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K-ary encoding and Hamming distance

• Redundant encoding
• Hamming distance is not natural.

1 1 2 2 2 3 3
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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.

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Labeling-independent Distance

• A 1213323, B 1122233

1 2 1 3 3 2 3
LI(A,B) 3
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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.

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Fitness Landscape Analysis
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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
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Normalized correlogram
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Normalized correlogram
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Global Convexity
Hamming distance
Correlation coefficient -0.11
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Global Convexity
Labeling-independent distance
Correlation coefficient 0.79
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Experimental Results
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Genetic Framework

• GA FM variant
• Population size 50
• Selection
• Roulette-wheel proportional selection
• Replacement
• Genitor-style replacement
• Steady-state GA

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Test Environment

• Data Set
• Johnson