Alg lgebraic Crossover Operators for Permutations Valentino Santucci 1,2 , Marco Baioletti 2 , Alfredo Milani 2 1 University for Foreigners of Perugia, Italy 2 University of Perugia, Italy
Finitely Generated Group β’ A group is a set π endowed with an operation β: π Γ π β π that: β’ is associative π¦ β π§ β π¨ = π¦ β π§ β π¨ βπ¦, π§, π¨ β π β’ has a neutral element π β π s.t. π¦ β π = π β π¦ = π¦ βπ¦ β π β π¦ β1 s.t. π¦ β π¦ β1 = π¦ β1 β π¦ = π β’ has inverse elements βπ¦ β π β’ A group π is finitely generated if there exists a finite generating set πΌ β π such that every π¦ β π can be expressed as a finite composition of the generators in πΌ , i.e., π¦ = β π 1 β β π 2 β β― β β π π with β β β πΌ β’ The length of a minimal decomposition of π¦ in terms of H is the weight of π¦ , we denote it with |π¦| β’ π¦ β π§ iff there exists (at least) a minimal decomposition of x that is a prefix of a minimal decomposition of y 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 2
Permutations form a group β’ A permutation of [π] = {1,2, β¦ , π} is a bijective discrete function from [π] to [π] , thus it is possible to compose permutations: π¨ = π¦ β π§ iff π¨ π = π¦(π§ π ) for 1 β€ π β€ π β’ The composition: β’ is associative π¦ β π§ β π¨ = π¦ β (π§ β π¨) β’ has neutral element π = 1,2, β¦ , π π¦ β1 π = π iff π¦ π = π β’ has inverse elements β’ The permutations of [π] , together with the β operation, form a group structure called the symmetric group π―(π) 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 3
Permutations form a F.G. group β’ Adjacent swap moves as generating set π΅ππ = π 1 , π 2 , β¦ , π πβ1 where, for any 1 β€ π < π : π + 1 if π = π π if π = π + 1 π π π = ΰ΅ π otherwise β’ (π¦ β π π ) is the permutation π¦ where the items at positions π and π + 1 have been swapped 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 4
Cayley graph β’ A finitely generated group induces a colored digraph (namely, a Cayley graph) where: β’ π are the vertices β’ there exists an arc π¦ β (π¦ β β) colored by β for every π¦ β π and β β πΌ β π β’ Connection between a Cayley graph and a combinatorial search space: β’ π is the set of solutions β’ πΌ β π represents simple search moves in the space of solutions β’ The Cayley graph induces: β’ neighborhood relationships among solutions β’ a distance between solutions (shortest path distance) β’ a single representation for both solutions and displacements 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 5
Cayley graph Generators: <2134> <1324> <1243> 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 6
Vector-like operations β’ Paths on the Cayley graph can be encoded by composing the labels on the arcs => paths can be encoded using permutations! β’ Permutations encode both Β«pointsΒ» and Β«vectorsΒ» in the search space β’ Letβs define β, β, β in such a way that they work consistently w.r.t. their usual numerical counterparts! 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 7
Sum and Difference of Permutations β’ Given two n -length permutations x and y : β’ π¦ β π§ β π¦ β π§ β’ π¦ β π§ β π§ β1 β π¦ β’ They are geometrically meaningful: β’ π¦ β π§ is the point reachable starting from point x and following the path y β’ π¦ β π§ represents a path connecting the point y to the point x β’ They are algebraically consistent: π¦ = π§ β π¦ β π§ = π§ β π§ β1 β π¦ = π¦ 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 8
Multiply a permutation by a scalar in [0,1] β’ Given a n -length permutation z and a scalar π β 0,1 β’ Letβs assume: β’ z represents a path β’ π¨ = π π 1 β π π 2 β β― β π π π is a minimal decomposition of π¨ with length π β’ π β π¨ β π π 1 β π π 2 β β― β π π π where π = π β π β’ Geometrically, π β π¨ is a Β«truncationΒ» of the path z 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 9
How to compute them? β’ β and β simply require permutations composition and inversion β’ β requires an algorithm for computing minimal decomposition(s) β’ There can be multiple minimal decompositions β’ They can be computed using a Β«bubble sortΒ»-like algorithm: Iteratively choose (and apply) an adjacent swap moving the permutation closer to the identity permutation (the only sorted permutation) β’ Two different strategies: β’ RandBS: randomly choose one suitable adjacent swap β’ GreedyBS: choose the best (and suitable) adjacent swap basing on the fitness function 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 10
RandBS and GreedyBS 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 11
What can we do with them? β’ Algebraic Differential Evolution β’ Algebraic Particle Swarm Optimization β’ Discretize numerical EAs whose Β«move rulesΒ» are linear combinations of solutions β’ β¦ design an algebraic crossover 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 12
Algebraic Crossovers β’ We propose 3 classes of algebraic crossovers: β’ Group-based algebraic crossovers β’ Lattice-based algebraic crossovers β’ Hybrid algebraic crossovers 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 13
Group-based Algebraic Crossovers (AXG) β’ Reasonably, a crossover between two permutations x and y should return a permutation z that is Β«in the middleΒ» between x and y β’ The interval [ x,y ] can be formally defined in many equivalent ways: β’ A group-based algebraic crossover operator AXG can be abstractly defined as any operator which, given two permutations x and y , returns a permutation z = AXG( x,y ) such that π¨ β [π¦, π§] 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 14
A property on pairwise precedences of items β’ z = AXG( x , y ) β’ AXG is precedence-respectful: z contains all the common precedences between x and y β’ AXG transmits precedences: all the common precedences of z come from x or y (no new precedence is generated) 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 15
How to compute AXG? β’ Ideally: β’ π¨ = π¦ β π β (π§ β π¦) , with π β [0,1] β’ Different strategies basing on the value of a and the β computation strategy β’ Practically: β’ We enumerate all the permutations in a shortest path from x to y by running RandBS (R) or GreedyBS (G) on π§ β π¦ β’ We select one permutation in the path: a random one (R), the middle one (T), the best one (B) β’ 6 different implementations: β’ AXG-RR, AXG-RT, AXG-RB β’ AXG-GR, AXG-GT, AXG-GB 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 16
The partial order β is a Lattice β’ Meet: π¨ = π¦ β§ π§ iff 1. π¨ β π¦ and π¨ β π§ 2. z is the Β«longestΒ» permutation with prop.1 β’ Join: π¨ = π¦ β¨ π§ iff 1. π¦ β π¨ and π§ β π¨ 2. z is the Β«shortestΒ» permutation with prop.1 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 17
Meet and Join as Crossover Operators AXL-Join exploits the Β«De MorganΒ»-like property: 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 18
Hybrid Algebraic Crossovers β’ Let AXG be any group-based algebraic crossover β’ Let m = AXL-Meet( x,y ) β’ Let j = AXL-Join( x,y ) β’ An hybrid algebraic crossover AXH is defined as AXH( x , y ) := AXG( m,j ) β’ 6 hybrid alg. crossovers: one for each AXG crossover 11 Jul 2018 - WCCI 2018 - Rio de Janeiro (Brasil) Algebraic Crossover Operators for Permutations - V. Santucci, M. Baioletti, A. Milani 19
Recommend
More recommend
