1. Geometric Crossover for Sets, Multisets and Partitions Alberto Moraglio and Riccardo Poli Abstract We extend a geometric framework for interpreting crossover to the case of sets and related representations. We also show that a deep geometric duality exists between the set representation and the vector representation that reveals the equivalence of geometric crossovers for these representations. Geometric Crossover It is based on the notion of metric 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 xy Traditional Crossover The traditional 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) Distances for Sets Symmetric distance: Ins/del edit distance: is the minimum number of elements that need to be inserted or deleted to transform one set into the other Theorem: Symmetric distance = Ins/del distance Corollary: both distances are metrics Example: A={a,b}, B={b,c} Symmetric: AΔB={a,c}  d(A,B)=2 Edit: from A delete a, insert c to obtain B  d(A,B)=2 Geometric Crossover for Sets Theorem: Given two parents sets A and B any recombination operator that returns offspring O such as is geometric crossover under symmetric distance Example: A={a,b}, B={b,c} If O={a,b,c}, d(A,O)=1, d(O,B)=1, d(A,B)=2  in the segment Distances & Crossover for Multisets Multiset: each element can occur more than once Operators: inclusion, union, intersection and symmetric difference for multisets are simple generalization of those for sets Distance: the symmetric distance is the size of the symmetric difference of two multisets, it is a metric and it is equivalent to the ins/del edit distance based on edit operations applied a single occurrence of an element Crossover: the geometric crossover associated to this symmetric distance is the same as the geometric crossover for simple sets using operators for multisets Distances & Crossover for Partitions Partition: a partition of a set X is a division of X into non-overlapping subsets that cover all of X Set as a bipartition: a n-partition is a generalization of a set seen as a bipartition of the universal set into the set and its complementary Edit distance: generalization of ins/del edit distance for partitions. It is based on moving an element from a subset to another. This move preserves mutual exclusivity of subsets and full coverage of the partition. Symmetric distance and edit distance for partitions do not coincide Crossover: the geometric crossover associated with this edit distance requires the offspring partition to satisfy with the constraint that the subsets need to form a partition (mutually exclusive and exhaustive cover) Set-Vector Duality Geometric crossovers based on isometric spaces are equivalent. The space of sets endowed with the symmetric distance is isometric to the space of vectors endowed with the Hamming distance through the indicator function of the set. Hence symmetric crossover for sets is equivalent to traditional crossover for vectors Example: Fixed-Size Sets Fixed-size sets: as further requirement we consider sets all of the same size Edit distance: minimum number of elements that need to be swapped among the two sets to transform one into another. This edit move never changes the size of the sets Crossover: the geometric crossover under this edit distance is the same as the variable-size sets case with the further requirement that offspring must have the same size of the parents