Abstract
Ant Colony Optimization (ACO) has become quite popular in recent years. In contrast to many successful applications, the theoretical foundation of this randomized search heuristic is rather weak. Building up such a theory is demanded to understand how these heuristics work as well as to come up with better algorithms for certain problems. Up to now, only convergence results have been achieved showing that optimal solutions can be obtained in finite time. We present the first runtime analysis of an ACO algorithm, which transfers many rigorous results with respect to the runtime of a simple evolutionary algorithm to our algorithm. Moreover, we examine the choice of the evaporation factor, a crucial parameter in ACO algorithms, in detail. By deriving new lower bounds on the tails of sums of independent Poisson trials, we determine the effect of the evaporation factor almost completely and prove a phase transition from exponential to polynomial runtime.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Dorigo, M., Blum, C.: Ant colony optimization theory: A survey. Theor. Comput. Sci. 344, 243–278 (2005)
Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT, Cambridge (2004)
Dorigo, M., Maniezzo, V., Colorni, A.: The ant system: An autocatalytic optimizing process. Tech. Rep. 91-016 Revised, Politecnico di Milano, Italy (1991)
Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)
Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edn., vol. 1. Wiley, New York (1968)
Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd edn., vol. 2. Wiley, New York (1971)
Giel, O., Wegener, I.: Evolutionary algorithms and the maximum matching problem. In: Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 2607, pp. 415–426. Springer, Berlin (2003)
Gleser, L.J.: On the distribution of the number of successes in independent trials. Ann. Probab. 3(1), 182–188 (1975)
Gutjahr, W.J.: A generalized convergence result for the graph-based ant system metaheuristic. Probab. Eng. Inf. Sci. 17, 545–569 (2003)
Gutjahr, W.J.: On the finite-time dynamics of ant colony optimization. Methodol. Comput. Appl. Probab. 8, 105–133 (2006)
Jerrum, M., Sorkin, G.B.: The Metropolis algorithm for graph bisection. Discrete Appl. Math. 82(1–3), 155–175 (1998)
Neumann, F., Wegener, I.: Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO ’04). Lecture Notes in Computer Science, vol. 3102, pp. 713–724. Springer, Berlin (2004)
Papadimitriou, C.H., Schäffer, A.A., Yannakakis, M.: On the complexity of local search. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (STOC ’90), pp. 438–445. ACM Press, Cambridge (1990)
Scheideler, C.: Probabilistic methods for coordination problems. HNI-Verlagsschriftenreihe 78. Habilitation Thesis, University of Paderborn. Available at http://www14.in.tum.de/personen/scheideler/index.html.en (2000)
Wegener, I.: Simulated annealing beats Metropolis in combinatorial optimization. In: Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP ’05). Lecture Notes in Computer Science, vol. 3580, pp. 589–601. Springer, Berlin (2005)
Witt, C.: Worst-case and average-case approximations by simple randomized search heuristics. In: Proceedings of the 22nd Annual Symposium on Theoretical Aspects of Computer Science (STACS ’05). Lecture Notes in Computer Science, vol. 3404, pp. 44–56. Springer, Berlin (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this paper appeared in the Proceedings of the 17th International Symposium on Algorithms and Computation (ISAAC 2006), volume 4288 of LNCS, pp. 618–627, Springer.
Financial support for C. Witt by the Deutsche Forschungsgemeinschaft (SFB) in terms of the Collaborative Research Center “Computational Intelligence” (SFB 531) is gratefully acknowledged.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Neumann, F., Witt, C. Runtime Analysis of a Simple Ant Colony Optimization Algorithm. Algorithmica 54, 243–255 (2009). https://doi.org/10.1007/s00453-007-9134-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-007-9134-2