Abstract
The paper provides some theoretical results on the analysis of the expected time needed by a class of Ant Colony Optimization algorithms to solve combinatorial optimization problems. A part of the study refers to some general results on the expected runtime of the considered class of algorithms. These results are then specialized to the case of pseudo-Boolean functions. In particular, three well known functions and a combination of two of them are considered: the OneMax, the Needle-in-a-Haystack, the LeadingOnes, and the OneMax-Needle-in-a-Haystack. The results obtained for these functions are also compared to those from the well-investigated (1+1)-Evolutionary Algorithm. The results shed light on a suitable parameter choice for the considered class of algorithms. Furthermore, it turns out that for two of the four studied problems, the expected runtime for the considered class, expressed in terms of the problem size, is of the same order as that for (1+1)-Evolutionary Algorithm. For the other two problems, the results are significantly in favour of the considered class of Ant Colony Optimization algorithms.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Birattari, P. Pellegrini, and M. Dorigo, “On the invariance of ant colony optimization,” Res. Rept, TR/IRIDIA/2006-025, 2006.
P. A. Borisovsky and A. V. Eremeev, “A study on performance of the (1+1)-evolutionary algorithm,” Proceedings on Foundations of Genetic Algorithms 7, Morgan Kaufmann: San Francisco, 2003.
M. Dorigo and C. Blum, “Ant colony optimization theory: a survey,” Theoretical Computer Science vol. 344 pp. 243–278, 2005.
M. Dorigo and G. Di Caro, “The ant colony optimization metaheuristic.” In D. Corne, M. Dorigo, and F. Glover (eds.), New Ideas in Optimization, pp. 11–32, McGraw-Hill: New York, 1999.
M. Dorigo, V. Maniezzo, and A. Colorni, “The ant system: an autocatalytic optimization process,” Res. Rept, pp. 91–016, Dept. of Electronics, Politecnico di Milano, Italy, 1991.
M. Dorigo, V. Maniezzo, and A. Colorni, “The ant system: optimization by a colony of cooperating agents,” IEEE Transactions On Systems, Man, and Cybernetics vol. 26 pp. 1–13, 1996.
M. Dorigo and T. Stützle, Ant Colony Optimization, MIT Press: Cambridge, 2004.
S. Droste, T. Jansen, and I. Wegener, “On the analysis of the (1+1) evolutionary algorithm,” Theoretical Computer Science vol. 276 pp. 51–81, 2002.
W. J. Gutjahr, “A graph–based ant system and its convergence,” Future Generations Computer Systems vol. 16 pp. 873–888, 2000.
W. J. Gutjahr, “ACO algorithms with guaranteed convergence to the optimal solution,” Information Processing Letters vol. 82 pp. 145–153, 2002.
W. J. Gutjahr, “A generalized convergence result for the graph–based ant system,” Probability in the Engineering and Informational Sciences vol. 17 pp. 545–569, 2003.
W. J. Gutjahr, “Theory of ant colony optimization: status and perspectives.” In MIC ’05 (6th Metaheuristics International Conference), Proceedings CD-ROM, 2005.
W. J. Gutjahr, “On the finite-time dynamics of ant colony optimization,” Methodology and Computing in Applied Probability vol. 8 pp. 105–133, 2006.
W. J. Gutjahr, “First steps to the runtime complexity analysis of ant colony optimization,” Computers & Operations Research 2007, click article in press at http://www.sciencedirect.com/science/journal/03050548
W. J. Gutjahr and G. Sebastiani, “Runtime analysis of ant colony optimization,” Res. Rept, 2007-03, Department of Mathematics, “Sapienza” University of Rome, 2007, available at www.mat.uniroma1.it/people/sebastiani/preprints.htm.
T. Jones and S. Forrest, “Fitness distance correlation as a measure of problem difficulty for genetic algorithms.” In Proc. 6th Int. Conf. on Genetic Algorithms, pp. 184–192, Morgan Kaufmann: San Mateo, 1995.
L. Kallel, B. Naudts, and C. R. Reeves, “Properties of fitness functions and search landscapes.” In L. Kallel, B. Naudts, and A. Rogers (eds.), Theoretical Aspects of Evolutionary Computing, pp. 174–206, Springer: Berlin, Heidelberg, New York, 1998.
V. Ladret, “Asymptotic hitting times for a simple evolutionary model of protein folding,” Journal of Applied Probability vol. 42 pp. 39–51, 2005.
D. Merkle and M. Middendorf, “Modeling the dynamics of ant colony optimization,” Evolutionary Computation vol. 10 pp. 235–262, 2002.
F. Neumann and C. Witt, “Runtime analysis of a simple ant colony optimization algorithm,” Res. Rept, CI-200/06, Dept. of Computer Science, University of Dortmund, Germany, 2006.
G. Sebastiani and G. L. Torrisi, “An extended ant colony algorithm and its convergence analysis,” Methodology and Computing in Applied Probability vol. 7 pp. 249–263, 2005.
T. Stützle and M. Dorigo, “A short convergence proof for a class of ACO algorithms,” IEEE Transactions on Evolutionary Computation vol. 6 pp. 358–365, 2002.
T. Stützle and H. H. Hoos, “The MAX-MIN Ant System and local search for the travelling salesman problem.” In T. Baeck, Z. Michalewicz, and X. Yao (eds.), Proc. ICEC ’97 Int. Conf. on Evolutionary Computation, pp. 309–314, 1997.
T. Stützle and H. H. Hoos, “MAX-MIN Ant System,” Future Generations Computer Systems vol. 16 pp. 889–914, 2000.
I. Wegener and C. Witt, “On the analysis of a simple evolutionary algorithm on quadratic pseudo-Boolean functions,” Journal of Discrete Algorithms vol. 3 pp. 61–78, 2005.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gutjahr, W.J., Sebastiani, G. Runtime Analysis of Ant Colony Optimization with Best-So-Far Reinforcement. Methodol Comput Appl Probab 10, 409–433 (2008). https://doi.org/10.1007/s11009-007-9047-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-007-9047-1