Geometric Crossover for Permutations with Repetitions:

Application to Graph Partitioning

A. Moraglio, Y-H. Kim, Y. Yoon, B-R. Moon & R. Poli

PPSN 2006

Contents

I. Geometric CrossoverII. Geometric Crossover for Permutation with

RepetitionsIII. Geometric Crossover for Graph Partitioning IV. Combination with Labelling-Independent CrossoverV. Experimental ResultsVI. Conclusions

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)

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!

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!

Geometric Crossover for Permutations with Repetitions

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

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

Generalized Cycle Crossover - Phase 1: find cycles

Generalized Cycle Crossover - Phase 2: mix cycles

Properties of the New Crossover

1. it preserves repetition class 2. it is a proper generalization of the cycle

crossover (when applied to simple permutations, it behaves exactly like the cycle crossover)

3. it searches only a fraction of the space searched by traditional crossover

4. when applied to parent permutations with repetitions of different repetition class, offspring have intermediate repetition class

Geometric Crossover for Multiway Graph Partitioning

Cut size : 5

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7

1 1 2 2 2 3 3

Cut size : 6

1

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7

1 3 1 2 2 3 2

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

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

Combination withLabelling-Independent Crossover

Graph encoding and Hamming distance

• Redundant encoding– Hamming distance is not natural.

1

2

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7

35

1 1 2 2 2 3 3

2233311

3311122

2211133

1122233

1133322

6 different representations

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.

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

Experimental Results

Crossovers

Experimental Results

32-way partitioning (average results)

0

1

2

3

4

5

6

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

5pt H-GX Cycle H-GX 5pt LI-GX Cycle LI-GX

Experimental Results

32-way partitioning (average results)

0

2

4

6

8

10

12

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

5pt H-GX Cycle H-GX 5pt LI-GX Cycle LI-GX

Experimental Results

128-way partitioning (average results)

0

0.5

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1.5

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2.5

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3.5

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

5pt H-GX Cycle H-GX 5pt LI-GX Cycle LI-GX

Experimental Results

128-way partitioning (average results)

00.20.40.60.8

11.21.41.61.8

2

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

5pt H-GX Cycle H-GX 5pt LI-GX Cycle LI-GX

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!

Thanks for your attention…questions?