Kernels of Mallows Models for Solving Permutation-based Problems

Josu Ceberio, Alexander Mendiburu, Jose A. Lozano

Genetic and Evolutionary Computation Conference (GECCO 2015)Madrid, Spain, 11-15 July 2015

Intelligent Systems GroupDepartment of Computer Science and Artificial Intelligence

University of the Basque Country UPV/EHU

2

Preliminaries

3

Combinatorial optimization problems

Permutation optimization problems

4

Permutation optimization problems

Problems whose solutions are naturally

represented as permutations

Travelling Salesman Problem (TSP)

1

2

6

3 5

4

8

7

Recently…A new trend of Estimation of Distribution Algorithms for permutation problems have been proposed

EDAs quicklyInitialize populationWhile stopping criterion is not met

Selected most promising individualsLearn a probability distributionSample new individuals from the distributionUpdate the population

Return the best solution

Mallows

6

The Mallows modelDefinition

• A distance-based exponential probability model

• Central permutation

• Spread parameter

• A distance on permutations

7

The Mallows modelDefinition

• A distance-based exponential probability model

• Central permutation

• Spread parameter

• A distance on permutations

8

The Mallows modelDefinition

• A distance-based exponential probability model

• Central permutation

• Spread parameter

• A distance on permutations

The Mallows modelKendall’s-τ distance

9

Measures the number of pairwise disagreements between and .

1-21-31-41-52-32-42-53-43-54-5

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The Mallows modelCayley distance

Measures the minimum number of swap operations to convert in .

The Mallows modelLearning

11

5 4 1 2 34 2 3 5 11 2 3 5 42 4 3 5 13 1 4 5 22 3 4 1 52 3 4 5 12 5 4 3 11 2 5 4 35 3 1 2 4

Population

Calculate the central permutation- Borda (Kendall’s-τ)- Set median permutation (Cayley)

Estimate the spread parameter - Newton-Raphson

12

The Mallows modelSampling

• For both metrics, the distance between and can be decomposed as a sum of terms.

Factorized distribution

Drawbacks

13

These models are unimodal

Often too restrictive

Drawbacks

14

What happens if good fitness solutions are far from each other?

Many works in the literature have proposed using mixtures of models

Drawbacks

15

Many works in the literature have proposed using mixtures of models

Building mixture models implies

costly algorithms

What happens if good fitness solutions are far from each other?

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The proposal:Kernels of Mallows models

Kernels of Mallows models

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5 4 1 2 34 2 3 5 11 2 3 5 42 4 3 5 13 1 4 5 22 3 4 1 52 3 4 5 12 5 4 3 11 2 5 4 35 3 1 2 4

Population

Mallows Kernels EDA

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Initialize population with individualsInitialize and

While stopping criterion is not metSelected the most promising individualsDefine Mallows kernels from the selected individualsSample individuals from each kernelUpdate populationUpdate

Return the best solution

There is no learning step

Repeated solutions are discardedExploration / Explotation

trade-off

19

Experimental Study

The question

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Can EDAs based on Kernels of Mallows Models outperformMallows EDA (MEDA) or Generalized Mallows EDA (GMEDA)?

Experimental Design

21

• Algorithms:– Mallows EDA (M)– Generalized Mallows EDA (GM)– Mallows Kernel EDA (K)

• Distances:- Kendall’s-τ- Cayley

Benchmark Problems:- Quadratic Assignment Problem (QAP)- Permutation Flowshop Scheduling Problem (PFSP)

22

1

2

3

4

5

67

8

12

34

5678

Experimental DesignThe quadratic assignment problem (QAP)

23

1

2

3

4

5

67

8

12

34

5678

Experimental DesignThe quadratic assignment problem (QAP)

Experimental DesignPermutation Flowshop Scheduling Problem (PFSP)

Total flow time (TFT)

m1m2m3m4

j4j1 j3j2 j5

• jobs• machines • processing times

5 x 4

24

Experimental DesignInstances

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• 45 Artificial QAP instances:– Size: 100, 150, 200, 250, 300, 350, 400,

450, 500– 5 instances of each size

• 45 Artificial PFSP instances:– Jobs: 50, 100, 200, 250, 300, 350, 400,

450, 500– Machines: 20– 5 instances of each size

Sampling uniformly at random from

[1,100]

Sampling parameters from Taillard’s

instances: tai80a ,tai80b,

tai100a,…

Experimental Design

26

• Parameter Settings:– Population size: 10n– Selected individuals: n– Selection type: truncation

– and :

0.00 2.00 4.00 6.00 8.00 10.00 12.000.000.100.200.300.400.500.600.700.800.901.00

Kendall’s-τ distance

θ

P(σ0

)

– Sampled individuals: 10n-1– Stopping criterion: 1000n2 evals– 10 repetitions

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.000.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Cayley distance

θ

P(σ0

)

100 150 200 250 300 350 400 450 5000.000.020.040.060.080.100.120.140.160.18

Cayley, QAP GMMK

Instance size

50 100 200 250 300 350 400 450 5000.040.050.060.070.080.090.100.110.120.13

Kendall, PFSP

Instance size

Experimental StudyARPD Results

50 100 200 250 300 350 400 450 5000.00

0.01

0.02

0.03

0.04Cayley, PFSP

Instance size

100 150 200 250 300 350 400 450 5000.10

0.15

0.20

0.25

0.30

0.35Kendall, QAP

Instance size

50 100 200 250 300 350 400 450 5000.00

0.01

0.02

0.03

0.04Cayley, PFSP

Instance size

50 100 200 250 300 350 400 450 5000.040.050.060.070.080.090.100.110.120.13

Kendall, PFSP

Instance size100 150 200 250 300 350 400 450 500

0.10

0.15

0.20

0.25

0.30

0.35Kendall, QAP

Instance size

100 150 200 250 300 350 400 450 5000.000.020.040.060.080.100.120.140.160.18

Cayley, QAP GMMK

Instance size

Experimental StudyARPD Results

Statistical AnalysisTwo non-parametric Friedman tests (α=0.05)

• QAP (6 algorithms):• PFSP (6 algorithms):

p-value < 0.001p-value < 0.001

50 100 200 250 300 350 400 450 5000.00

0.01

0.02

0.03

0.04Cayley, PFSP

Instance size

50 100 200 250 300 350 400 450 5000.040.050.060.070.080.090.100.110.120.13

Kendall, PFSP

Instance size100 150 200 250 300 350 400 450 500

0.10

0.15

0.20

0.25

0.30

0.35Kendall, QAP

Instance size

100 150 200 250 300 350 400 450 5000.000.020.040.060.080.100.120.140.160.18

Cayley, QAP GMMK

Instance size

Experimental StudyARPD Results

Statistical AnalysisTwo non-parametric Friedman tests (α=0.05)

• QAP (6 algorithms):• PFSP (6 algorithms):

Shaffer’s static posthoc – pairwise comparisons

• QAP:

• PFSP:

p-value < 0.001p-value < 0.001

50 100 200 250 300 350 400 450 5000.0E+00

2.0E+05

4.0E+05

6.0E+05

8.0E+05

1.0E+06

1.2E+06

1.4E+06

1.6E+06

1.8E+06Kendall, PFSP

Instance size100 150 200 250 300 350 400 450 500

0.0E+002.0E+054.0E+056.0E+058.0E+051.0E+061.2E+061.4E+061.6E+061.8E+06

Kendall, QAP

Instance size

100 150 200 250 300 350 400 450 5000.0E+002.0E+044.0E+046.0E+048.0E+041.0E+051.2E+051.4E+051.6E+05

Cayley, QAPGMMK

Instance size50 100 200 250 300 350 400 450 500

0.0E+00

1.0E+04

2.0E+04

3.0E+04

4.0E+04

5.0E+04

6.0E+04Cayley, PFSP

Instance size

Experimental StudyComputational Cost (seconds)

Conclusions

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Can EDAs based on Kernels of Mallows Models outperformMallows EDA (MEDA) and Generalized Mallows EDA (GMEDA)?

Yes, if they are defined under the Cayley distance

In fact, Kernels of Mallows models under the Cayley distance are the best option in terms

of fitness and computational cost

Future Work

32

Short termUnderstand why similar results are not obtained with Kendall’s-τ

Apply automatic (offline) algorithm configuration: Irace

Study more advanced strategies to adapt θs

Long term

Include other distances such as Ulam or Hamming

Apply to other permutation problems

Thank you for your

attention!