1. Geometric Crossover for Permutations with Repetitions: Application to Graph Partitioning A. Moraglio, Y-H. Kim, Y. Yoon, B-R. Moon & R. Poli PPSN 2006

2. Contents I.Geometric Crossover II.Geometric Crossover for Permutation with Repetitions III.Geometric Crossover for Graph Partitioning IV.Combination with Labelling-Independent Crossover V.Experimental Results VI.Conclusions

3. Geometric Crossover

4. 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

5. 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)

6. 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!

7. 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!

8. Geometric Crossover for Permutations with Repetitions

9. 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

10. 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

11. Generalized Cycle Crossover - Phase 1: find cycles

12. Generalized Cycle Crossover - Phase 2: mix cycles

13. 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

14. Geometric Crossover for Multiway Graph Partitioning

15. Cut size : 5 1 2 3 4 5 6 7 1122233

16. Cut size : 6 1 2 3 4 5 6 7 1312232

17. 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

18. 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

19. Combination with Labelling-Independent Crossover

20. Graph 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

21. 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.

22. 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

23. Experimental Results

24. Crossovers

25. Experimental Results 32-way partitioning (average results)

26. Experimental Results 32-way partitioning (average results)

27. Experimental Results 128-way partitioning (average results)

28. Experimental Results 128-way partitioning (average results)

29. 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!

30. Thanks for your attention… questions?