320 likes | 603 Views
PPSN 2006. Geometric Crossover for Permutations with Repetitions: Application to Graph Partitioning. A. Moraglio, Y-H. Kim, Y. Yoon, B-R. Moon & R. Poli. Contents. Geometric Crossover Geometric Crossover for Permutation with Repetitions Geometric Crossover for Graph Partitioning
E N D
PPSN 2006 Geometric Crossover for Permutations with Repetitions:Application to Graph Partitioning A. Moraglio, Y-H. Kim, Y. Yoon, B-R. Moon & R. Poli
Contents • Geometric Crossover • Geometric Crossover for Permutation with Repetitions • Geometric Crossover for Graph Partitioning • Combination with Labelling-Independent Crossover • Experimental Results • Conclusions
y x 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
B 1 0 1 1 0 1 X A 2 B 1 1 0 1 1 3 A X 0 1 0 1 1 Geometric Crossover • The traditional n-point crossover is geometric under the Hamming distance. 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 • 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
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
2 6 1 4 7 3 5 Cut size : 5
6 2 1 4 7 3 5 Cut size : 6
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
6 2 1 4 1 1 3 3 3 2 2 7 2 2 1 1 1 3 3 3 2 2 3 3 3 1 1 5 3 3 1 1 1 2 2 6 different representations 3 3 2 2 2 1 1 Graph encoding and Hamming distance • Redundant encoding • Hamming distance is not natural.
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 32-way partitioning (average results)
Experimental Results 32-way partitioning (average results)
Experimental Results 128-way partitioning (average results)
Experimental Results 128-way partitioning (average results)
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!