Geometric Crossover for the Permutation Representation

1. GSICE 2005 Geometric Crossover for the Permutation Representation Alberto Moraglio & Riccardo Poli {amoragn,rpoli}@essex.ac.uk

2. Contents • Abstract Geometric Operators • Geometric Crossover for Permutations • Geometric Crossover for TSP • Conclusions

3. I. Abstract Geometric Operators

4. Binary Strings Permutations 100000011101000 100110011101000 100111100011100 100001100011100 Crossover Is there any common aspect ? Is it possible to give a representation- independent definition of crossover and mutation? Real Vectors Syntactic Trees What is crossover?

5. Mutation & Nearness • Mutation is naturally interpreted in terms of nearness: offspring are near the parent • Example: Binary StringP = 0 1 0 1 11O = 0 1 0 1 0 1 • NEARNESS:hd(P,O)=1

6. Crossover & Betweenness • Crossover is naturally interpreted in terms of betweenness: offspring are between parents • Example: Binary StringP1 = 0 1 0|0 1 0P2 = 1 1 0|1 0 1O = 0 1 0 1 0 1hd(P1,P2)=4hd(P1,O)=3 hd(O,P2)=1 • BETWEENNES: P1---O-P2

7. 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

8. 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)

9. 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

10. 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

11. II. Geometric Crossover Design for Permutations

12. 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

13. 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

14. 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 BC D E F)  (A C B D E F) Edit Distance = minimum number of edit moves to transform one permutation into the other

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

16. MAGIC OF EDIT DISTANCES: Neighbourhood/syntax DUALITY • NEIGHBOURHOOD: Picking offspring on shortest path connecting two nodes • SYNTAX: picking offspring onminimal sorting trajectory between parent permutations using the edit move as sort move (minimal sorting by x)

17. Many sorting algorithms do minimal sorting by X Geometric Crossovers = Sorting Crossovers!

18. III. Geometric Crossover Design for TSP

19. 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

20. 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)

22. 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

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

24. 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

25. IV. Conclusions

26. 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!

27. Thank you for your attention… Questions?