Topological Crossover for the Permutation Representation

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GECCO 2005
Topological Crossover for the Permutation
Representation
Alberto Moraglio Riccardo Poli amoragn,rpoli_at_e
ssex.ac.uk
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Topological Crossover Abstract Geometric
Crossover
Sorry Name Change!
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Contents

1. Abstract Geometric Operators
2. Geometric Crossover for Permutations
3. Geometric Crossover for TSP
4. Conclusions

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I. Abstract Geometric Operators
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What is crossover?
Binary Strings
Permutations
Real Vectors
Syntactic Trees
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Shortest Path Crossover
Hamming Neighbourhood Structure
Parent1 011101 Parent2 010111 Children
0111
Crossover in the Neighbourhood offspring between
parents Mask-based crossover children are on
shortest paths
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From graphs to geometry

• Neighbourhood StructureMetric 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

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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.

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Squared balls Chunky segments
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Uniform Mutation Uniform Crossover

• Uniform topological crossover
• Uniform topological e-mutation

Genetic operators have a geometric nature
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Representation-independentand rigorous
definition ofcrossover and mutation in the
neighbourhood seen as a geometric space
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So what? Claims at Gecco 2004

1. EAs Unification most pre-existing genetic
   operators for main representations are geometric
2. Simplification Clarification crossover as
   function of classical neighbourhood structure
   simplifies the established notion of crossover
   landscape (hyper-neighbourhood) as function of
   crossover
3. General theory formal representation-independent
   definitions allow for a general theory
4. Crossover principled design specifying the
   formal definition of crossover for a specific
   representation and distance one gets
   automatically a specific crossover

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II. Geometric Crossover for Permutations
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Many Distances Dilemma
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Many Distances Dilemma
Representation Binary Strings Permutations
Distance One distance Hamming distance Many distances
Geometric Crossover Mask-based crossover Many types of crossover
Geometric Uniform Crossover Uniform crossover Many uniform crossovers

• WHAT IS A GOOD DISTANCE?
• WHAT IS THE RIGTH CROSSOVER?

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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

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Crossover Implementation Edit Distances
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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
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PermutationEdit Move Neighbourhood Structure
Shortest path distance edit distance
Line segment in the neighbourhood structure
all shortest paths connecting two nodes
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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)

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Many sorting algorithms do minimal sorting by X
Ordinary Sorting Algorithm Minimal Sorting by X
Bubble Sort Adj. Swap
Insertion Sort Insert
Selection Sort Swap
Quick Sort No Fix Move!
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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

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EmbeddingProblem Knowledge
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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

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N-queens - mutations
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N-queens - crossovers
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Crossover Rank vs. Mutation Rank
1. Selection Sort Uniform 1. Swap
2. PMX -
3. Selection Sort 1-point 1. Swap
4. Insertion Sort Uniform 2. Insertion
5. Insertion Sort 1-point 2. Insertion
6. Bubble Sort Uniform 3. Adj. Swap
7. Bubble Sort 1-point 3. Adj. Swap
Good mutation, good crossover heuristic
holds! Uniform crossovers are better than 1-point
crossovers
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III. Geometric Crossover for TSP
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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)

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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

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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
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• Until Population Convergence
• No Mutation
• Runs30 (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.

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Results for eil51 (small)
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Results for lin105 (medium)
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Results for kroA200 (medium-big)
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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

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IV. Conclusions
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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!

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Thank you for your attention Questions?
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N-queens - parameters
Problem size 100
Population size 5000
Mutation probability 0.1 (0)
Crossover probability (0) 1
Generation 500
Selection tournament size 5
Statistics Average 30 runs