O-BEE-COL

O-BEE-COL: Optimal BEEs for COLoring Graphs

Piero Consoli 1 Mario Pavone 2

1School of Computer Science,University of Birmingham

Edgbaston, Birmingham, B15 2TT, [email protected]

2Department of Mathematics and Computer Science,University of Catania

Viale A. Doria 6, 95125 Catania, [email protected]

http://www.dmi.unict.it/mpavone/

International Conference on Artificial Evolution - EA 2013

Mario Pavone (University of Catania) http://www.dmi.unict.it/mpavone/ [email protected] 1 / 26

Outline

1 Introduction & Aims

2 Graph Coloring ProblemDefinition and FormalizationProperty: lower and upper bounds

3 O–BEE–COL: Optimal BEEs for COLoring

4 ResultsParameters TuningImpact of the New OperatorsRunning Time, ttt–plotsResults & Comparisons

5 Conclusions


Introduction & Aims

Swarm Intelligence: collective intelligent behavior to solvecomplex problemsStrengths: self-organization & no centralized supervisionArtificial Bee Colony – ABC: inspired by the intelligent foragingbehavior of a colony of bees

I competitive in continuous optimization tasks[Karaboga et al., J. of Global Optimization, 39(3), 2007]

Graph Coloring Problem used for evaluate and compareAims:

1 evaluate performances in term of quality of solutionsF minimal number of colors found; average number of colors found;

success rate; and average evaluations number to solution2 impact factor of variants and operators designed3 search capabilities, efficiency and robustness


Graph Coloring Problem – GCP

Classical combinatorial optimization problem that findsapplicability in many real-world problemsLabelling/coloring the countries of a map such that no twoadjacent countries share the same label/colorG = (V ,E) =⇒ c : V → S such that c(u) 6= c(v) for any (u, v) ∈ Eif |S| = k then G is said k-colorableChromatic Number (χ): minimal cardinality of S

I G is said k-chromatic.

Computing χ is NP–complete problem


Chromatic Number Property

Two different approaches:I assignment of the colors to verticesI partitioning V into k groups: every group is an independent set

(NP–Complete, as well)

Theorem of Brooks∗: for any connected graph G

χ(G) ≤ ∆(G)

I if G is either a complete graph, or a odd cycle graph

χ(G) ≤ ∆(G) + 1,

χ(G) ≥ ω(G), where ω(G) is a maximum clique of GI [clique: a complete subgraph]

∗Brooks, proc. Cambridge Philosophical Society, Math. Phys. Sci., 37:194–197, 1941


Optimal BEEs for COLoring

O–BEE–COL


O-BEE-COL

Three different types of bees each with different taskEmployed Bees: search for food, and store information on foodsources

I tries to improve each solution using perturbation operatorsOnlooker Bees: exploits the information in order to select goodfood sources

I choose the solution to exploit with a roulette wheel selectionI number of onlooker bees is proportional to its fitness

Scout Bees: discover new food sourcesI all solutions without improvements are replaced by new ones

Strengths of O–BEE–COL via three operators: mutation,crossover and temperature mechanism


Creation Initial Population

Random ran-RLF ∗ Mixed

Instance k0 kf k̂ k0 kf k̂ kf

le450_15a 20 19 21.6266 17 17 17.7825 16le450_15b 19 19 21.5611 17 17 17.6643 16le450_15c 28 20 30.5457 24 24 25.9417 15le450_15d 28 18 30.6810 24 24 26.0832 15Dsjc125.1 7 6 8.1119 6 5 6.3719 6Dsjc125.5 22 18 25.3306 21 17 23.3237 17Dsjc125.9 51 44 56.0681 50 44 55.3191 44Dsjc250.1 11 10 12.8497 10 9 10.3916 9Dsjc250.5 38 30 42.5141 37 29 39.9272 29Dsjc250.9 90 74 97.6878 88 74 96.7181 73flat300_20 42 20 46.3438 40 20 43.6366 20flat300_26 43 29 47.1527 41 26 44.4959 26flat300_28 43 33 47.1430 41 32 44.4480 31

∗ ran-RLF: [Costa & Hertz, Journal of Operational Research Society, 48:295–305, 1997]


O-BEE-COL

SmartSwap mutation operator: tries to reduce the number ofcolorclasses handling the troublesome nodes

I δ: maximum number of constraints unsatisfied allowed

GPX – Greedy Partitioning Crossover: [Jin-Kao Hao et al., J. of

Combinatorial Optimization, 3(4):379–397, 1999]

optimized GPX: the cardinality of the color classes that can becopied are handled by a parameter

I generated conflicts are eliminatedDifferences:

1 more than 2 solutions are involved (partSol)2 cardinality of classes to be copied is determined by partLimit .


O-BEE-COL

Temperature mechanism: dynamically handles some parametersduring the evolutionary cycle

I number of parents involved in GPX =⇒ partSolI number of the improvement trails needed to replace a solution

=⇒ evLimitI number of scout bees =⇒ nScoutsI percentage of solutions generated by randomized RLF =⇒ percSol

if we inhibit one of them the outcome will be negatively affected


Results & Comparisons


Experimental Protocol

The challenging DIMACS Benchmark

http://mat.gsia.cmu.edu/COLOR/instances.html

10 indenpendent runsStop Criterion: maximum fitness function evaluations allowed

instance DSJC250.5popSize {200, 500, 1000, 1500, 2000}percEmp {10%, 20%, 50%, 70%, 90%}partLimit {5, 10, 15, 18}


Analysis of the Convergence Speed

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0 500 1000 1500 2000

k

generations

Artificial Bee Colony - popSize parameter

200,10%,5

500,10%,5

1000,10%,5

1500,10%,5

2000,10%,5

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0 500 1000 1500 2000

k

generations

Artificial Bee Colony - percEmp parameter

200,10%,5

200,20%,5

200,50%,5

200,70%,5

200,90%,5

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39

0 500 1000 1500 2000

k

generations

Artificial Bee Colony - partLimit parameter

200,10%,5

200,10%,10

200,10%,15

200,10%,18


All Operator Combinations for O-BEE-COL

variant SmartSwap Crossover Temperature k̂ k SR AES1 on opt GPX on 15 15 100% 5, 972, 9252 on GPX on 24 24 100% 1, 503, 7563 on opt GPX off 17.8 15 40% 36, 599, 0354 on GPX off 25 25 100% 55 off opt GPX on 15.9 15 50% 25, 981, 4206 off GPX on 24 24 100% 1, 639, 4037 off opt GPX off 19.9 17 20% 15, 872, 8348 off GPX off 25 25 100% 4


Impact Factor of SmartSwap Operator

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0 25000 50000 75000 100000 125000 150000

k

generations

1 best5 best

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0 25000 50000 75000 100000 125000 150000

k

generations

3 best7 best

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0 25000 50000 75000 100000 125000 150000

k

generations

2 best6 best


Impact Factor of Optimized GPX

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0 25000 50000 75000 100000 125000 150000

k

generations

1 best2 best

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0 25000 50000 75000 100000 125000 150000

k

generations

7 best8 best

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0 25000 50000 75000 100000 125000 150000

k

generations

3 best4 best

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0 25000 50000 75000 100000 125000 150000

k

generations

5 best6 best


Impact Factor of Temperature mechanism

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0 25000 50000 75000 100000 125000 150000

k

generations

1 best3 best

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0 25000 50000 75000 100000 125000 150000

k

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5 best7 best

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0 25000 50000 75000 100000 125000 150000

k

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2 best4 best


Time-To-Target plots

Standard graphical methodology for data analysis [Chambers et al., Chapman & Hall,

1983]

A way to characterize the running time of stochastic algorithms

Display the probability that an algorithm will find a solution as good as atarget within a given running time

a Perl program – tttplots.pl – to create time-to-target plots[Resende et al., Optimization Letters, 2007] – [http://www2.research.att.com/˜mgcr/tttplots/]

Experiments on instances where the obtained mean is equal to theoptimal solution (SR = 100%)

Analysis conducted:

I School1 and DSJC250.1I 200 independent runsI termination criterion: until finding the target solution

Larger is the number of runs closer is the empirical distribution to thetheoretical distribution


ttt-plots

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time to target solution

Artificial Bee Colony

empirical

theoretical 0

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theoretical

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

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exponential quantiles


empirical

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+1 std dev range

-1 std dev range 0

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+1 std dev range

-1 std dev range


DIMACS Benchmark Instances

Graph | V | | E | χ k∗ k SR AES

DSJC125.1 125 736 5 5 5 50% 528, 715.6DSJC125.5 125 3, 891 12 17 17 10% 464, 633.0DSJC125.9 125 6, 961 30 42 44 100% 29, 817.4DSJC250.1 250 3, 218 8 8 9 100% 252, 538.7DSJC250.5 250 15, 668 13 28 29 100% 471, 823.0DSJC250.9 250 27, 897 35 69 73 90% 24, 403, 325.4le450_15a 450 8, 168 15 15 16 100% 17, 678, 139.9le450_15b 450 8, 169 15 15 16 100% 6, 188, 035.6le450_15c 450 16, 680 15 15 15 100% 5, 972, 925.6le450_15d 450 16, 750 15 15 15 80% 18, 630, 401.3flat300_20 300 21, 375 20 20 20 100% 4, 800flat300_26 300 21, 633 26 26 26 100% 72.9Kflat300_28 300 21, 695 28 28 31 20% 5.6MQueen5_5 25 320 5 5 5 100% 1.9Queen6_6 36 580 7 7 7 100% 1, 741.66Queen7_7 49 952 7 7 7 100% 6, 636.84Queen8_8 64 1, 456 9 9 9 100% 22, 107.25Queen8_12 96 2, 736 12 12 12 100% 1, 212, 000.35Queen9_9 81 1, 056 10 10 10 100% 31, 243.28School1.nsh 352 14, 612 14 14 14 100% 1, 703.28School1 385 19, 095 14 14 14 100% 821.5


O–BEE–COL vs. Other Algorithms

instance I_Greedy T_B&B DIST PAR T_GEN_1 O–BEE–COL

DSJC125.5 18.9 20.0 17.0 17.0 17.0 17.9DSJC250.5 32.8 35.0 28.0 29.2 29.0 29.0flat300_20 20.0 39.0 20.0 20.0 20.0 20.0flat300_26 37.1 41.0 26.0 32.4 26.0 26.0flat300_28 37.0 41.0 31.0 33.0 33.0 31.8le450_15a 17.9 16.0 15.0 15.0 15.0 16.0le450_15b 17.9 15.0 15.0 15.0 15.0 16.0le450_15c 25.6 23.0 15 16.6 16.0 15.0le450_15d 25.8 23.0 15.0 16.8 16.0 15.2mulsol.i.1 49.0 49.0 49.0 49.0 49.0 49.0school1.nsh 14.1 26.0 20.0 14.0 14.0 14.0

I_Greedy: [Culberson et al., 2nd DIMACS Implementation Challenge, 14(1):254–284, 1996]

T_B&B: [Glover et al., 2nd DIMACS Implementation Challenge, 14(1):285–307, 1996]

DIST: [Morgenstern, 2nd DIMACS Implementation Challenge, 14(1):335–357, 1996]

PAR: [Lewandowski et al., 2nd DIMACS Implementation Challenge, 14(1):309–334, 1996]

T_GEN_1: [Fleurent et al., 2nd DIMACS Implementation Challenge, 14(1):619–652, 1996]



instance IMMALG ANTCOL EVOLVE_AO MACOL O–BEE–COL AS-GCP

DSJC125.5 18.0 18.7 17.2 17.0 17.9 17.0DSJC250.5 28.0 31.0 29.1 28.0 29.0 29.7flat300_20 20.0 20.0 26.0 20.0 20.0 20.0flat300_26 27.0 34.4 31.0 26.0 26.0 32.6flat300_28 32.0 34.3 33.0 29.0 31.8 32.6le450_15a 15.0 16.0 15.0 15.0 16.0 16.0le450_15b 15.0 16.0 15.0 15.0 16.0 16.0le450_15c 15.0 15.0 16.0 15.0 15.0 15.0le450_15d 16.0 15.0 19.0 15.0 15.2 15.0mulsol.i.1 49.0 - - - 49.0 49.0school1.nsh 15.0 - - 14.0 14.0 14.0

IMMALG: [Cutello et al., Journal of Combinatorial Optimization, 14(1):9–33, 2007] & [Pavone et al., Journal of GlobalOptimization, 53(4):769–808, 2012]

ANTCOL: [Costa et al., Journal of Operational Research Society, 49:295–305, 1997]

EVOLVE_AO: [Barbosa et al., Journal of Combinatorial Optimization, 8(1):41–63, 2004]

MACOL: [Hao et al., European Journal of Operational Research, 203(1):241–250, 2010]

AS-GCP: [Pavone et al., IEEE Congress on Evolutionary Computation, 1:1909–1916, 2013]



Graph O-BEE-COL HPSO HCA GPB VNS VSS HANTCOL

DSJC250.5 29 28 28 28 - - 28flat300_26 26 26 - - 31 - -flat300_28 31 31 31 31 31 29 31le450_15c 15 15 15 15 15 15 15le450_15d 15 15 - - 15 15 -

The results have been taken from [Qin et al., Journal of Computers, 6(6):1175–1182, 2011] except HCA

HPSO: [Qin et al., Journal of Computers, 6(6):1175–1182, 2011]

HCA: [Hao et al., Journal of Combinatorial Optimization, 3(4):379–397, 1999]

GPB: [Glass et al., Journal of Combinatorial Optimization, 7(3):229–236, 2003]

VNS: [Hertz et al., European Journal of Operational Research, 151(2):379–388, 2003]

VSS: [Hertz et al., Discrete Applied Mathematics, 156(13):2551–2560, 2008]

HANTCOL: [Thompson et al., Discrete Applied Mathematics, 156(3):313–324, 2008]



Graph O-BEE-COL IGrAl FCNS IPM ABAC LAVCA TPA AMACOL

DSJC125.1 5 5 5 6 5 5 5 5DSJC125.5 17 17 18 19 17 17 19 17DSJC125.9 44 43 44 45 44 44 44 44DSJC250.1 9 8 − 10 8 8 8 8DSJC250.5 29 29 − − 29 28 30 28DSJC250.9 73 72 − 75 72 72 72 72le450_15a 16 15 − − 15 15 15 15le450_15b 16 15 − 17 15 15 15 15le450_15c 15 16 − 17 15 15 15 15le450_15d 15 16 − − 15 15 15 15Queen5_5 5 5 − − 5 − − −Queen6_6 7 7 − − 7 − − −Queen7_7 7 7 − − 7 − − −Queen8_8 9 9 9 9 9 − − −Queen8_12 12 12 − − 12 − − −Queen9_9 10 10 10 10 10 − − −school1_nsh 14 14 − − 14 − − −School1 14 14 − − 14 − − −

IGrAl: [Caramia et al., Discrete Applied Mathematics, 156:201–217, 2008]

FCNS: [Prestwich, Discrete Applied Mathematics, 156:148–158, 2008]

IPM: [Dukanovic et al., Discrete Applied Mathematics, 156:180–189, 2008]

ABAC: [Bui et al., Discrete Applied Mathematics, 156:190–200, 2008]

LAVCA, TPA, & AMACOL: [Torkestain et al., Computing and Informatics, 29(1):447–466, 2010]


Conclusions

A comparative analysis is presented on the designed Artificial BeeColony (O–BEE–COL)

O–BEE–COL: based on (1) SmartSwap mutation operator; (2) optimizedGPX; and (3) Temperature mechanism

GCP has been tackled in order to evaluate the performances

Many experiments have been performed in order to:I find the best tuning of the parametersI analyze the running time via Time-To-Target plotsI evaluate the impact factor of the variants and operators introducedI performed experimental comparisons

All possible combinations of the three operators have been taken intoaccount

I inhibiting one of them the performances are negatively affected

Many comparisons with other algorithms have been conducted

O–BEE–COL showed efficiency; robustness; and competitiveperformances


Thanks!


O-BEE-COL

Technology

connected graph g g

labelcolor g

odd cycle graph g g

optimal bees

partsol number

number of colorclasses

evlimit number of scout

tnessscout bees