Topological Crossover for the Permutation Representation Alberto Moraglio & Riccardo Poli GECCO 2005

Topological Crossover Abstract Geometric Crossover Sorry… Name Change!

Contents I.Abstract Geometric Operators II.Geometric Crossover for Permutations III.Geometric Crossover for TSP IV.Conclusions

I. Abstract Geometric Operators

What is crossover? Crossover Is there any common aspect ? Is it possible to give a representation- independent definition of crossover and mutation? Binary Strings Permutations Real Vectors Syntactic Trees

Shortest Path Crossover D0 : P1 D2 : P2 D1 Parent1: Parent2: Children: 01*1*1 Crossover in the Neighbourhood: offspring between parents Mask-based crossover: children are on shortest paths Hamming Neighbourhood Structure

From graphs to geometry Neighbourhood Structure=Metric Space The distance in the neighbourhood is the length of the shortest path connecting two solutions Mutation  Direct neighbourhood  Ball Crossover  All shortest paths  Line Segment

Balls & Segments In a metric space (S, d) the closed ball is the set of the form where x belongs to S and r is a positive real number called the radius of the ball. In a metric space (S, d) the line segment or closed interval is the set of the form where x and y belong to S and are called extremes of the segment and identify the segment.

Squared balls & Chunky segments B(000; 1) Hamming space 3 B((3, 3); 1) Euclidean space 3 B((3, 3); 1) Manhattan space Balls [000; 011] = [001; 010] 2 geodesics Hamming space 1 3 [(1, 1); (3, 2)] 1 geodesic Euclidean space 1 3 [(1, 1); (3, 2)] = [(1, 2); (3, 1)] infinitely many geodesics Manhattan space Line segments

Uniform Mutation & Uniform Crossover Uniform topological crossover: Uniform topological ε-mutation: Genetic operators have a geometric nature

Representation-independent and rigorous definition of crossover and mutation in the neighbourhood seen as a geometric space

So what? Claims at Gecco 2004 (i)EAs Unification: most pre-existing genetic operators for main representations are geometric (ii)Simplification & Clarification: crossover as function of classical neighbourhood structure simplifies the established notion of crossover landscape (hyper- neighbourhood) as function of crossover (iii)General theory: formal representation-independent definitions allow for a general theory (iv)Crossover principled design: specifying the formal definition of crossover for a specific representation and distance one gets automatically a specific crossover

II. Geometric Crossover for Permutations

Many Distances Dilemma

WHAT IS A GOOD DISTANCE? WHAT IS THE RIGTH CROSSOVER? RepresentationBinary StringsPermutations DistanceOne distance =Hamming distanceMany distances Geometric CrossoverMask-based crossoverMany types of crossover Geometric Uniform CrossoverUniform crossoverMany uniform crossovers

What is a good distance? –IN PRINCIPLE: abstract genetic operators are well- defined for any distance. However: –IMPLEMENTATION: a distance not rooted in the solution syntax does not tell how to implement crossover –PROBLEM KNOWLEDGE: a problem-independent distance does not put any problem knowledge in the search –A GOOD DISTANCE: –(i) suggests how to implement crossover –(ii) embeds problem knowledge in the algorithm

Crossover Implementation & Edit Distances

Mutations/Edit moves for Permutations Reversal: (A B C D E F)  (A E D C B F) Insert: (A B C D E F)  (A C D E B F) Swap: (A B C D E F)  (A D C B E F) Adj.Swap: (A B C D E F)  (A C B D E F) Edit Distance = minimum number of edit moves to transform one permutation into the other

Permutation+Edit Move = Neighbourhood Structure Shortest path distance = edit distance abc bac acb bca cab cba B(abc; 1) Adjacent swap space abc bac acb bca cab cba [abc; bca] 1 geodesic Adjacent swap space B(abc; 1) Swap space & Reversal space abc bac acb bca cab cba abc bac acb bca cab cba [abc; bca] 3 geodesics Swap space & Reversal space B(abc; 1) Insertion space [abc; bca] 1 geodesic Insertion space abc bac acb bca cab cba abc bac acb bca cab cba Line segment in the neighbourhood structure = all shortest paths connecting two nodes

Neighbourhood/syntax duality NEIGHBOURHOOD: Picking offspring on shortest path connecting two nodes SYNTAX: picking offspring on minimal sorting trajectory between parent permutations using the edit move as sort move (minimal sorting by x)

Many sorting algorithms do minimal sorting by X Ordinary Sorting Algorithm Minimal Sorting by X Bubble SortAdj. Swap Insertion SortInsert Selection SortSwap Quick SortNo Fix Move!

Geometric Crossovers = Sorting Crossovers! Sorting Crossover by X: –sorting one parent permutation toward the other using X sort move –stop the sorting at random and return the partially sorted permutation as offspring Bubble Sort Crossover = Geometric Crossover under adj. swap edit distance

Embedding Problem Knowledge

Edit Distances & Problem Knowledge How can we pick an edit distance that embeds problem knowledge? Minimal fitness change: pick the edit distance whose edit move corresponds to a minimal fitness change Good mutation, Good crossover: pick the edit distance whose edit move corresponds to a good mutation for the problem at hand Good neighbourhood, Good crossover: pick the edit distance whose edit move induces a neighbourhood structure that is known to be good for the problem

N-queens - mutations

N-queens - crossovers

Crossover Rank vs. Mutation Rank 1. Selection Sort Uniform1. Swap 2. PMX- 3. Selection Sort 1-point1. Swap 4. Insertion Sort Uniform2. Insertion 5. Insertion Sort 1-point2. Insertion 6. Bubble Sort Uniform3. Adj. Swap 7. Bubble Sort 1-point3. Adj. Swap Good mutation, good crossover heuristic holds! Uniform crossovers are better than 1-point crossovers

III. Geometric Crossover for TSP

Geometric Crossover for TSP A good neighbourhood structure for TSP is 2opt structure = space of circular permutations endowed with reversal edit distance Geometric crossover for TSP = picking offspring on the minimal sorting trajectories by sorting one parent circular permutation toward the other parent by reversals (sorting circular permutations by reversals)

Approximated Geometric Crossover BAD NEWS: sorting circular permutations by reversals is NP-Hard! GOOD NEWS: there are approximation algorithms that sort within a bounded error to optimality (used in genetics) A 2-approximation algorithm sorts by reversals using sorting trajectories that are at most twice the length of the minimal sorting trajectories Approximation algorithms can be used to build approximated geometric crossovers for TSP

Experiments - Parameters Test-bed TSPLIB: eil51, gr96, eil101, lin105, d198, kroA200, lin318, pcb442 Crossovers PMX: partially matched crossover ERX: edge recombination SBRX: sorting by reversal crossover (limitations: no circular permutation, uniform on one fixed geodesic, 2-approxiamtion) Parameter Setting BIG POPULATION: Population Size = Instance Size * 20 Until Population Convergence No Mutation Runs=30 (average of bests in population) No Fine Tuning. The settings have been chosen to allow the best crossover to reach a near optimal solution before convergence.

Results for eil51 (small)

Results for lin105 (medium)

Results for kroA200 (medium-big)

Good results & lot of room for improvement SBRX better than ERX for bigger instances good empirical results based only on theoretical considerations Possible improvements: –Fine parameter tuning –Better approximation algorithm –Non-deterministic approx algorithm (uniform crossover) –Circular Permutations instead of Linear Permutations

IV. Conclusions

Conclusions Permutations & Many Distances –Many types of geometric crossovers! –What is a good distance? Implementation & Edit Distance: –Edit Distances are good –For permutations: geometric crossovers = sorting algorithms! Problem Knowledge and Edit Move: –Good mutation, good crossover heuristics –For permutations: good mutation, good crossover holds for the N- queen problem using sorting crossovers Geometric Crossover for TSP –Sorting circular permutation by reversals (NP-Hard) –2-approximation algorithm for approximated geometric crossover –Good empirical results based only on theory!

Thank you for your attention… Questions?

N-queens - parameters Problem size100 Population size5000 Mutation probability0.1 (0) Crossover probability(0) 1 Generation500 Selectiontournament size 5 StatisticsAverage 30 runs