Swarm Intelligence Heuristics for Graph Coloring Problem
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This document presents two novel swarm intelligence heuristics, AS-GCP (Ant System based Graph Coloring Problem) and ABC-GCP (Artificial Bee Colony based Graph Coloring Problem), aimed at efficiently solving the graph coloring problem, a well-known NP-complete issue. It evaluates the performance, solution quality, and efficiency of these methods through experimental comparisons and results, demonstrating their effectiveness against established benchmarks. The study primarily concludes which of the two heuristics is more suitable for coloring a graph based on various tuning parameters and performance metrics.
![Introduction & Aims
Introduction & Aims
Two novel Swarm Heuristics: AS-GCP and ABC-GCP
Ant Systems: most efficient and robust of the family
Artificial Bee Colony: competitive in optimization tasks
[Karaboga et al., J. of Global Optimization, 39(3), 2007]
Graph Coloring Problem used for evaluate and compare
Aims:
1 evaluate performances of AS-GCP and ABC-GCP
2 quality of solutions: minimal colors number found; average colors
number found; success rate; and average evaluations number to
solution
3 impact factor of variants and novelties designed
4 search capabilities
5 efficiency and robustness
6 Which of the two is more suitable for coloring a graph
Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 3 / 24](https://image.slidesharecdn.com/ccp-swarm-cec2013-130629052326-phpapp01/85/Swarm-Intelligence-Heuristics-for-Graph-Coloring-Problem-3-320.jpg)
![Graph Coloring Problem
Graph Coloring Problem – GCP
Classical combinatorial optimization problem that finds
applicability in many real-world problems
Given G = (V, E) assign one color C such that C(u) = C(v) for
any (u, v) ∈ E
Partitioning V into groups: every group is an independent set
Chromatic Number (χ): minimal number of colors
if |C| = k then G is said k-colorable
if k = χ G is said k-chromatic.
Computing χ is NP–complete problem [Garey and Johnson,1979]
Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 4 / 24](https://image.slidesharecdn.com/ccp-swarm-cec2013-130629052326-phpapp01/85/Swarm-Intelligence-Heuristics-for-Graph-Coloring-Problem-4-320.jpg)
![Swarm Intelligence Heuristics AS-GCP: an Ant Colony System
AS-GCP
AS-GCP is based on ANTCOL [Costa et al., J. of Oper. Research Society, 49:295–305, 1997]
Randomized ANT_RLF: combines the good heuristic of RLF with the
pheromone trails
Pi,k =
τα
ik ηβ
ik
j∈W τα
ij ηβ
ij
Fitness function: takes account the number of colors, and length of each
color class
f(x) =
1
||V| − |Ck || · c(x)
Pheromone trail:
τv,k =
1, if Ck is empty
u∈Ck
Mu,v
|Ck | , otherwise.
Evaporation mechanism: Mr,s = (1 − ρ) Mr,s
Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 6 / 24](https://image.slidesharecdn.com/ccp-swarm-cec2013-130629052326-phpapp01/85/Swarm-Intelligence-Heuristics-for-Graph-Coloring-Problem-6-320.jpg)
![Swarm Intelligence Heuristics AS-GCP: an Ant Colony System
AS-GCP
Any ant undergoes to Crossover Operator and Local Search
Greedy Partitioning Crossover (GPX): most important the set of
the vertices that belong to the same class [Jin-Kao Hao et al., J. of
Combinatorial Optimization, 3(4): 379–397, 1997]
Local Search: recursively tries to decrease the number of colors
of each solution
Depends on the parameters: minRec and maxRec
Integration among GPX and Local Search: avoid fragmentation of
the colorclasses
refinement via LS if the cardinality is below to partLim
Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 7 / 24](https://image.slidesharecdn.com/ccp-swarm-cec2013-130629052326-phpapp01/85/Swarm-Intelligence-Heuristics-for-Graph-Coloring-Problem-7-320.jpg)
![Results Running Time – ttt-plots
Time-To-Target plots
Standard graphical methodology for data analysis
[Chambers et la., 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 a
target 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/]
Both swarm heuristics are run where the obtained mean is equal to the
optimal solution
Analysis conducted on: le450_15b and DSJC250.1 200 runs
termination criterion: until finding the target solution
larger is the number of runs closer is the empirical distribution to the
theoretical distribution
Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 17 / 24](https://image.slidesharecdn.com/ccp-swarm-cec2013-130629052326-phpapp01/85/Swarm-Intelligence-Heuristics-for-Graph-Coloring-Problem-23-320.jpg)
Swarm Intelligence Heuristics for Graph Coloring Problem
- 1. Swarm Intelligence Heuristics for Graph Coloring Problem Piero Consoli 1 Alessio Collerà 2 Mario Pavone 2 1School of Computer Science, University of Birmingham Edgbaston, Birmingham, B15 2TT, UK 2Department of Mathematics and Computer Science, University of Catania Viale A. Doria 6, 95125 Catania, Italy IEEE Congress on Evolutionary Computation – CEC 2013 Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 1 / 24
- 2. Outline 1 Introduction & Aims 2 Graph Coloring Problem 3 Swarm Intelligence Heuristics AS-GCP: an Ant Colony System ABC-GCP: an Artificial Bee Colony 4 Results Parameters Tuning Running Time – ttt-plots Experimental Comparisons 5 Conclusions Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 2 / 24
- 3. Introduction & Aims Introduction & Aims Two novel Swarm Heuristics: AS-GCP and ABC-GCP Ant Systems: most efficient and robust of the family Artificial Bee Colony: competitive in optimization tasks [Karaboga et al., J. of Global Optimization, 39(3), 2007] Graph Coloring Problem used for evaluate and compare Aims: 1 evaluate performances of AS-GCP and ABC-GCP 2 quality of solutions: minimal colors number found; average colors number found; success rate; and average evaluations number to solution 3 impact factor of variants and novelties designed 4 search capabilities 5 efficiency and robustness 6 Which of the two is more suitable for coloring a graph Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 3 / 24
- 4. Graph Coloring Problem Graph Coloring Problem – GCP Classical combinatorial optimization problem that finds applicability in many real-world problems Given G = (V, E) assign one color C such that C(u) = C(v) for any (u, v) ∈ E Partitioning V into groups: every group is an independent set Chromatic Number (χ): minimal number of colors if |C| = k then G is said k-colorable if k = χ G is said k-chromatic. Computing χ is NP–complete problem [Garey and Johnson,1979] Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 4 / 24
- 5. Swarm Intelligence Heuristics AS-GCP: an Ant Colony System an Ant Colony System – AS-GCP Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 5 / 24
- 6. Swarm Intelligence Heuristics AS-GCP: an Ant Colony System AS-GCP AS-GCP is based on ANTCOL [Costa et al., J. of Oper. Research Society, 49:295–305, 1997] Randomized ANT_RLF: combines the good heuristic of RLF with the pheromone trails Pi,k = τα ik ηβ ik j∈W τα ij ηβ ij Fitness function: takes account the number of colors, and length of each color class f(x) = 1 ||V| − |Ck || · c(x) Pheromone trail: τv,k = 1, if Ck is empty u∈Ck Mu,v |Ck | , otherwise. Evaporation mechanism: Mr,s = (1 − ρ) Mr,s Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 6 / 24
- 7. Swarm Intelligence Heuristics AS-GCP: an Ant Colony System AS-GCP Any ant undergoes to Crossover Operator and Local Search Greedy Partitioning Crossover (GPX): most important the set of the vertices that belong to the same class [Jin-Kao Hao et al., J. of Combinatorial Optimization, 3(4): 379–397, 1997] Local Search: recursively tries to decrease the number of colors of each solution Depends on the parameters: minRec and maxRec Integration among GPX and Local Search: avoid fragmentation of the colorclasses refinement via LS if the cardinality is below to partLim Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 7 / 24
- 8. Swarm Intelligence Heuristics AS-GCP: an Ant Colony System AS-GCP vs. ANTCOL ANTCOL AS-GCP instance k bf SR ± AES bf SR ± AES Queen6_6 7 7 100% ± 956 7 100%± 96 Queen7_7 7 7 61% ± 1991 7 100% ±700 Queen8_8 9 9 48% ± 1294 9 100% ±203 Queen8_12 12 12 87% ± 6625 12 100% ±162 Queen9_9 10 10 60% ± 2614 10 100% ±1022 School1.nsh 14 14 100% ± 2966 14 100%± 1417 School1 14 14 100% ± 3120 14 100%± 922 DSJC125.1 5 5 40% ± 9310 5 100% ±160 DSJC125.5 12 18 30% ± 9751 17 100% ± 4590 DSJC125.9 30 44 40% ± 10112 44 100% ± 1128 DSJC250.1 8 9 100% ± 2561 8 100% ± 21773 DSJC250.5 13 31 100% ± 19238 29 30% ± 24156 DSJC250.9 35 75 40% ± 9274 73 40% ± 7554 le450_15a 15 16 100% ± 7913 16 100%± 1600 le450_15b 15 16 100% ± 3872 16 100%± 1052 le450_15c 15 15 100% ± 12422 15 100%± 4300 le450_15d 15 15 100% ± 12755 15 100%± 8604 flat300_20 20 20 100% ± 8188 20 100%± 6394 flat300_26 26 34 60% ± 21548 32 60% ± 10780 flat300_28 28 34 70% ± 19120 32 60% ± 9186 Table: Improvements produced by novelties introduced. Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 8 / 24
- 9. Swarm Intelligence Heuristics ABC-GCP: an Artificial Bee Colony an Artificial Bee Colony – ABC-GCP Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 9 / 24
- 10. Swarm Intelligence Heuristics ABC-GCP: an Artificial Bee Colony ABC-GCP Three different types of bees each with different task Employed Bees: search for food, and store information on food sources tries to improve each solution using perturbation operators Onlooker Bees: exploits the information in order to select good food sources choose the solution to exploit with a roulette wheel selection Scout Bees: discover new food sources all solutions without improvements are replaced by new ones Initialization phase: RLF + random order Strengths via three operators: mutation, crossover and temperature mechanism Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 10 / 24
- 11. Swarm Intelligence Heuristics ABC-GCP: an Artificial Bee Colony ABC-GCP SmartSwap mutation operator: tries to reduce the number of colorclasses handling the troublesome nodes partLimit: maximum number of constraints unsatisfied allowed optimized GPX: the cardinality of the color classes that can be copied are handled by a parameter generated conflicts are eliminated Temperature mechanism: dynamically handles some parameters during the evolutionary cycle number of parents involved in GPX number of the improvement trails needed to replace a solution number of scout bees percentage of solutions generated by RLF during the scout bees phase if we inhibit one of them the outcome will be negatively affected Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 11 / 24
- 12. Swarm Intelligence Heuristics ABC-GCP: an Artificial Bee Colony Impact Factor of ABC-GCP Operators 12 13 45 44 46 5 47555 75555 87555155555147555175555 9
- 13. 8 12 13 45 44 46 5 47555 75555 87555155555147555175555 9
- 14. 8 3 12 13 45 44 46 5 47555 75555 87555155555147555175555 9
- 15. 7 8 12 13 45 44 46 5 47555 75555 87555 155555147555175555 9
- 16. 1 7 12 13 45 44 46 5 47555 75555 87555 155555147555175555 9
- 17. 1 4 12 13 45 44 46 5 47555 75555 87555155555147555175555 9
- 18. 1 Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 12 / 24
- 19. Results Results Comparisons Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 13 / 24
- 20. Results Experimental Protocolo Aims The challenging DIMACS Benchmark http://mat.gsia.cmu.edu/COLOR/instances.html 10 indenpendent runs Stop Criterion: maximum fitness function evaluations allowed AS-GCP ABC-GCP le450_15c DSJC250.5 α ∈ {0, 1, 2, 3, 4} popSize ∈ {200, 500, 1000, 1500, 2000} β ∈ {0, 1, 2, 3, 4} percEmp ∈ {10%, 20%, 50%, 70%, 90%} ρ ∈ {0.01, 0.1, 0.3, 0.6, 0.9, 0.99} partLimit ∈ {5, 10, 15, 18} minRec ∈ {1, 5, 10, 30} maxRec ∈ {5, 10, 30} partLim ∈ {1, 3, 5, 8} Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 14 / 24
- 21. Results Parameters Tuning Analysis of the Convergence Speed: AS-GCP 15 20 25 30 35 0 5 10 15 20 k generations Ant Colony Optimization - alpha parameter 2,4,0.99,5,10,5 0,4,0.99,5,10,5 1,4,0.99,5,10,5 3,4,0.99,5,10,5 4,4,0.99,5,10,5 15 20 25 30 35 0 25 50 75 100 k generations Ant Colony Optimization - beta parameter 2,4,0.99,5,10,5 2,0,0.99,5,10,5 2,1,0.99,5,10,5 2,2,0.99,5,10,5 2,3,0.99,5,10,5 15 20 25 30 35 0 25 50 75 100 k generations Ant Colony Optimization - rho parameter 2,4,0.99,5,10,5 2,4,0.01,5,10,5 2,4,0.1,5,10,5 2,4,0.3,5,10,5 2,4,0.6,5,10,5 2,4,0.9,5,10,5 15 20 25 30 35 0 5 10 15 20 k generations Ant Colony Optimization - maxRec parameter 2,4,0.99,5,10,5 2,4,0.99,5,5,5 2,4,0.99,5,30,5 15 20 25 30 35 0 5 10 15 20 k generations Ant Colony Optimization - minRec parameter 2,4,0.99,5,10,5 2,4,0.99,1,10,5 2,4,0.99,10,10,5 15 20 25 30 35 0 5 10 15 20 k generations Ant Colony Optimization - partLim parameter 2,4,0.99,5,10,5 2,4,0.99,5,10,1 2,4,0.99,5,10,3 2,4,0.99,5,10,8 Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 15 / 24
- 22. Results Parameters Tuning Analysis of the Convergence Speed: ABC-GCP 29 30 31 32 33 34 35 36 37 38 39 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 29 30 31 32 33 34 35 36 37 38 39 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 29 30 31 32 33 34 35 36 37 38 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 Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 16 / 24
- 23. Results Running Time – ttt-plots Time-To-Target plots Standard graphical methodology for data analysis [Chambers et la., 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 a target 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/] Both swarm heuristics are run where the obtained mean is equal to the optimal solution Analysis conducted on: le450_15b and DSJC250.1 200 runs termination criterion: until finding the target solution larger is the number of runs closer is the empirical distribution to the theoretical distribution Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 17 / 24
- 24. Results Running Time – ttt-plots ttt-plots 0 0.2 0.4 0.6 0.8 1 400 450 500 550 600 650 700 750 800 cumulativeprobability time to target solution Ant Colony System empirical theoretical 0 0.2 0.4 0.6 0.8 1 0 30 60 90 cumulativeprobability time to target solution Artificial Bee Colony empirical theoretical 500 550 600 650 700 750 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 measuredtimes exponential quantiles Ant Colony System empirical estimated +1 std dev range -1 std dev range 46 48 50 52 54 56 58 60 62 64 66 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 measuredtimes exponential quantiles Artificial Bee Colony empirical estimated +1 std dev range -1 std dev range Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 18 / 24
- 25. Results Experimental Comparisons AS-GCP vs. ABC-GCP ABC-GCP AS-GCP instance |V| |E| k bf SR AES bf SR AES Queen6_6 36 580 7 7 100% 1.7K 7 100% 96 Queen7_7 49 952 7 7 100% 6.6K 7 100% 700 Queen8_8 64 1,456 9 9 100% 22.1K 9 100% 203 Queen8_12 96 2,736 12 12 100% 1.2M 12 100% 162 Queen9_9 81 1,056 10 10 100% 31.2K 10 100% 1K School1.nsh 352 14,612 14 14 100% 1.7K 14 100% 1.4K School1 385 19,095 14 14 100% 821,5 14 100% 922 DSJC125.1 125 736 5 5 50% 528K 5 100% 160 DSJC125.5 125 3,891 12 17 10% 464K 17 100% 4.5K DSJC125.9 125 6,961 30 44 100% 29K 44 100% 1.1K DSJC250.1 250 3,218 8 9 100% 252K 8 100% 21.7K DSJC250.5 250 15,668 13 29 100% 471K 29 30% 24.5K DSJC250.9 250 27,897 35 73 90% 24.4M 73 40% 7.5K le450_15a 450 8,168 15 16 100% 17.6M 16 100% 1.6K le450_15b 450 8,169 15 16 100% 6.1M 16 100% 1K le450_15c 450 16,680 15 15 100% 5.9M 15 100% 4.3K le450_15d 450 16,750 15 15 80% 18.6M 15 100% 8.6K flat300_20 300 21,375 20 20 100% 4800 20 100% 6.3K flat300_26 300 21,633 26 26 100% 72.9K 32 60% 10.7K flat300_28 300 21,695 28 31 20% 5.6M 32 60% 9.1K Table: ABC-GCP versus AS-GCP on several DIMACS instances. Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 19 / 24
- 26. Results Experimental Comparisons AS-GCP ABC-GCP vs. Other Algorithms instance I_Greedy T_BB DIST PAR T_GEN_1 ABC-GCP AS-GCP DSJC125.5 18.9 20.0 17.0 17.0 17.0 17.9 17.0 DSJC250.5 32.8 35.0 28.0 29.2 29.0 29.0 29.7 flat300_20 20.0 39.0 20.0 20.0 20.0 20.0 20.0 flat300_26 37.1 41.0 26.0 32.4 26.0 26.0 32.6 flat300_28 37.0 41.0 31.0 33.0 33.0 31.8 32.6 le450_15a 17.9 16.0 15.0 15.0 15.0 16.0 16.0 le450_15b 17.9 15.0 15.0 15.0 15.0 16.0 16.0 le450_15c 25.6 23.0 15 16.6 16.0 15.0 15.0 le450_15d 25.8 23.0 15.0 16.8 16.0 15.2 15.0 mulsol.i.1 49.0 49.0 49.0 49.0 49.0 49.0 49.0 school1.nsh 14.1 26.0 20.0 14.0 14.0 14.0 14.0 I_Greedy: [Culberson et al., 2nd DIMACS Implementation Challenge, 14(1):254–284, 1996] T_BB: [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] Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 20 / 24
- 27. Results Experimental Comparisons AS-GCP ABC-GCP vs. Other Algorithms instance IMMALG ANTCOL EVOLVE_AO HCA MACOL ABC-GCP AS-GCP DSJC125.5 18.0 18.7 17.2 - 17.0 17.9 17.0 DSJC250.5 28.0 31.0 29.1 28.1 28.0 29.0 29.7 flat300_20 20.0 20.0 26.0 - 20.0 20.0 20.0 flat300_26 27.0 34.4 31.0 - 26.0 26.0 32.6 flat300_28 32.0 34.3 33.0 31.4 29.0 31.8 32.6 le450_15a 15.0 16.0 15.0 - 15.0 16.0 16.0 le450_15b 15.0 16.0 15.0 - 15.0 16.0 16.0 le450_15c 15.0 15.0 16.0 15.6 15.0 15.0 15.0 le450_15d 16.0 15.0 19.0 - 15.0 15.2 15.0 mulsol.i.1 49.0 - - - - 49.0 49.0 school1.nsh 15.0 - - - 14.0 14.0 14.0 IMMALG: [Cutello et al., Journal of Combinatorial Optimization, 14(1):9–33, 2007] and [Pavone et al., Journal of Global Optimization, 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] HCA: [Hao et al., Journal of Combinatorial Optimization, 3(4):379–397, 1999] MACOL: [Hao et al., European Journal of Operational Research, 203(1):241–250, 2010] Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 21 / 24
- 28. Results Experimental Comparisons AS-GCP ABC-GCP vs. Other Algorithms instance HPSO GPB VNS VSS HANTCOL ABC-GCP AS-GCP DSJC250.5 28 28 - - 28 29 29 flat300_26 26 - 31 - - 26 32 flat300_28 31 31 31 29 31 31 32 le450_15c 15 15 15 15 15 15 15 le450_15d 15 - 15 15 - 15 15 The results have been taken from [Qin et al., Journal of Computers, 6(6):1175–1182, 2011] HPSO: [Qin et al., Journal of Computers, 6(6):1175–1182, 2011] 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] Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 22 / 24
- 29. Conclusions Conclusions (1/2) A comparative study is presented between two swarm intelligence heuristics AS-GCP: based on the combination of GPX and LS ABC-GCP: based on SmartSwap mutation operator; optimized GPX; and Temperature mechanism GCP has been tackled in order to evaluate the performances Many experiments have been performed in order to: find the best tuning of the parameters analyze the running time via Time-To-Target plots performed experimental comparisons Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 23 / 24
- 30. Conclusions Conclusions (2/2) Inspecting ttt-plots: ABC-GCP produces good approximations of the theoretical distributions than AS-GCP Analyzing experiments: both swarm heuristics are competitive with all tested algorithms The efficiency and robustness showed proves us the goodness of the variants, and novelties designed AS-GCP vs. ABC-GCP: AS-GCP shows a quickly learning that helps in reaching the best coloring in a few generations ABC-GCP presents a behavior more robust and competitive than AS-GCP ABC-GCP seems to be more suitable on graphs with more dense, and complex topology Mario Pavone (mpavone@dmi.unict.it) http://www.dmi.unict.it/mpavone/ University of Catania 24 / 24