In this research work we present two novel swarm heuristics based respectively on the ants and bees artificial colonies, called AS-GCP and ABC-GCP. The first is based mainly on the combination of Greedy Partitioning Crossover (GPX), and a local search approach that interact with the pheromone trails system; the last, instead, has as strengths three evolutionary operators, such as a mutation operator; an improved version of GPX and a Temperature mechanism. The aim of this work is to evaluate the efficiency and robustness of both developed swarm heuristics, in order to solve the classical Graph Coloring Problem (GCP). Many experiments have been performed in order to study what is the real contribution of variants and novelty designed both in AS-GCP and ABC-GCP. A first study has been conducted with the purpose for setting the best parameters tuning, and analyze the running time for both algorithms. Once done that, both swarm heuristics have been compared with 15 different algorithms using the classical DIMACS benchmark. Inspecting all the experiments done is possible to say that AS-GCP and ABC-GCP are very competitive with all compared algorithms, demonstrating thus the goodness of the variants and novelty designed. Moreover, focusing only on the comparison among AS-GCP and ABC-GCP is possible to claim that, albeit both seem to be suitable to solve the GCP, they show different features: AS-GCP presents a quickly convergence towards good solutions, reaching often the best coloring; ABC-GCP, instead, shows performances more robust, mainly in graphs with a more dense, and complex topology. Finally, ABC-GCP in the overall has showed to be more competitive with all compared algorithms than AS-GCP as average of the best colors found.

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