Geometric Crossover for Permutations with Repetitions: Application to Graph Partitioning

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PPSN 2006
Geometric Crossover for Permutations with
RepetitionsApplication to Graph Partitioning
A. Moraglio, Y-H. Kim, Y. Yoon, B-R. Moon R.
Poli
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Contents

• Geometric Crossover
• Geometric Crossover for Permutation with
  Repetitions
• Geometric Crossover for Graph Partitioning
• Combination with Labelling-Independent Crossover
• Experimental Results
• Conclusions

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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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Many Recombinations are Geometric

• Traditional Crossover extended to multary strings
• Recombinations for real vectors
• PMX, Cycle Crossovers for permutations
• Homologous Crossover for GP trees
• Ask me for more examples over a coffee!

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Being geometric crossover is important because.

• We know how the search space is searched by
  geometric crossover for any representation
  convex search
• We know a rule-of-thumb on what type of
  landscapes geometric crossover will perform well
  smooth landscape
• This is just a beginning of general theory, in
  the future we will know more!

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Geometric Crossover for Permutations with
Repetitions
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Geometric Crossover for Permutations

• PMX geometric under swap distance
• Cycle Crossover geometric under swap and Hamming
  distance (restricted to permutations)
• More crossovers for permutations are geometric
• We extend Cycle Crossover to permutations with
  repetitions and show its application to the graph
  partitioning problem
• The extended Cycle Crossover is still geometric
  under Hamming distance (restricted to
  permutations with repetitions) but not geometric
  under swap distance

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Permutations with repetitions

• Simple permutation (21453)
• Permutation with repetitions
• (214151232)
• Repetition class (33111)
• We want to search the space of permutations
  belonging to the same repetition class

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Generalized Cycle Crossover - Phase 1 find cycles

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Generalized Cycle Crossover - Phase 2 mix cycles

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Properties of the New Crossover

• it preserves repetition class
• it is a proper generalization of the cycle
  crossover (when applied to simple permutations,
  it behaves exactly like the cycle crossover)
• it searches only a fraction of the space searched
  by traditional crossover
• when applied to parent permutations with
  repetitions of different repetition class,
  offspring have intermediate repetition class

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Geometric Crossover for Multiway Graph
Partitioning
15
2
6
1
4
7
3
5
Cut size 5
1 1 2 2 2 3 3
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Cut size 6
1 3 1 2 2 3 2
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Feasible Solutions

• Balanced Solution the difference in cardinality
  between the largest and the smallest subsets is
  at most one
• Balancedness is a hard constraint feasible
  solutions are balanced, infeasible solution are
  not balanced
• Our evolutionary algorithm does not use any
  repairing mechanism. It restricts the search to
  the space of the balanced solutions using search
  operators that preserve balancedness

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Searching Balanced Solution Space

• Representation permutation with repetitions.
  Each Position in the permutation corresponds to a
  vertex in the graph. Each element of the
  permutation corresponds to a group
• Initial Population equally balanced solutions
  belonging to the same repetition class
• Crossover cycle crossover that preserves
  repetition class, hence balancedness
• Mutation swap mutation that preserves repetition
  class, hence balancedness

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Combination withLabelling-Independent Crossover
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Graph 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 Crossover

• LI distance Minimum Hamming distance between
  partitions over all possible relabelling
• LI Geometric Crossover Relabel the second parent
  such as it is at minimum Hamming distance from
  first parent (normalization). Do the normal
  n-point crossover using the first parent and the
  normalized second parent.

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Combination of Cycle Crossover and
Labelling-Independent Crossover

• First normalization of second parent on first
  parent
• Then cycle crossover between first parent and
  normalised second parent
• Still geometric under LI-H distance restricted to
  balanced partitions

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Experimental Results
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Crossovers
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Experimental Results
32-way partitioning (average results)
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Experimental Results
32-way partitioning (average results)
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Experimental Results
128-way partitioning (average results)
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Experimental Results
128-way partitioning (average results)
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Summary

• Geometric crossover offspring are in the segment
  between parents
• Cycle crossover for permutation geometric under
  Hamming distance
• Generalized cycle crossover extension of cycle
  crossover with permutation with repetition. It is
  geometric under hamming distance and it is
  class-preserving
• Geometric crossover for graph partitioning it
  searches only the space of feasible solutions
  (balanced partitions) that is a fraction of the
  search space searched by traditional crossover
• Combination with labelling-independent crossover
  it filters the redundancy of the labelling and it
  searches only balanced partitions. It is a
  geometric crossover
• Experimental results the combined geometric
  crossover has remarkable performance!

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Thanks for your attentionquestions?