Geometric Crossover for the Permutation Representation Alberto Moraglio & Riccardo Poli GSICE 2005

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

Mutation & Nearness Mutation is naturally interpreted in terms of nearness: offspring are near the parent Example: Binary String P = O = NEARNESS:hd(P,O)=1

Crossover & Betweenness Crossover is naturally interpreted in terms of betweenness: offspring are between parents Example: Binary String P1 = 0 1 0|0 1 0 P2 = 1 1 0|1 0 1 O = hd(P1,P2)=4 hd(P1,O)=3 hd(O,P2)=1 BETWEENNES: P1---O-P2

Geometric Crossover DEFINITION: geometric crossover is any recombination operator for which there is at least a (metric) distance such as all offspring are between parents Definition properties: - is representation-independent -clear-cuts crossover from non-crossover -generalises many pre-existing crossovers

Geometric Crossovers across Representations Many pre-existing recombination operators are geometric under suitable distance: BINARY: one-point, two-points, uniform crossovers REAL VECTORS: line, arithmetic, discrete (non- geometric: extended line) PERMUTATIONS: PMX, Edge Recombination, Cycle Crossover, Merge Crossover (non-geometric: order crossover) SYNTACTIC TREES: homologous one-point & uniform crossovers (non-geometric: subtree swap crossover)

Geometric Operators Formalization BALL: All points within distance r from x SEGMENT: All points between x and y UNIFORM  -MUTATION: offspring z are taken uniformly within the ball of radius  from the parent x UNIFORM CROSSOVER: offspring z are taken uniformly within the segment between parents x and y

Advantages of Geometric Operators REPRESENTATION UNIFICATION: many pre-existing operators are geometric SIMPLIFIED ANALISYS: natural interpretation of crossover within the classic notion of neighbourhood & landscape GENERAL THEORY: formal definition + dynamical equations  representation- independent evolutionary dynamics CROSSOVER DESIGN: formal definition + specific distance  specific crossover

II. Geometric Crossover Design for Permutations

Distance & Representation IN PRINCIPLE: abstract genetic operators are well-defined for any distance without any reference to solution representation IMPLEMENTATION REQUIREMENT: however a distance must be rooted in the solution representation to make the crossover implementation possible (practical) EDIT DISTANCES: firmly rooted in the solution representation and guiding crossover implementation

One Representation, Many Crossovers Binary Strings are associated with Hamming Distance (HD) Uniform Geometric Crossover under HD corresponds to uniform crossover for binary strings Permutation representation can be naturally associated with many distances Since for each distance, there is one crossover: there are many different uniform geometric crossovers for permutation representation

Edit Distances 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

MAGIC OF EDIT DISTANCES: 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!

III. Geometric Crossover Design for TSP

Distance & Problem Knowledge IN PRINCIPLE: abstract genetic operators are well-defined for any distance without any reference to the problem at hand PROBLEM KNOWLEDGE REQUIREMENT: however, a problem-independent distance does not put any problem knowledge in the search. A good distance embeds problem knowledge. HEURISTICS: Good neighbourhood, Good crossover: pick the edit distance whose edit move induces a neighbourhood structure that is known to be good for the problem

Geometric Crossover for TSP A known 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

Results for TSPLIB (typical) Big Population – No mutation – Until Convergence

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 –Geometric uniform crossover –Circular permutations instead of linear permutations

IV. Conclusions

Summary Geometric Interpretation & Formalization of Genetic Operators: –Mutation  Nearness  Ball –Crossover  Betweenness  Line Segment Crossover Design for Permutations: –Implementation requirement: distance based on syntax –One representation, many distances  many crossovers –Edit distances for permutations: geometric crossovers = sorting algorithms! Crossover Design for TSP: –Problem knowledge requirement: distance makes landscape ‘smooth’ –Edit distance for TSP: reversal distance (2-opt) –Sorting circular permutations by reversals (NP-Hard) –2-approximation algorithm for approximated geometric crossover –Good empirical results based only on theory!

Thank you for your attention… Questions?