, 54:243 | Cite as

Runtime Analysis of a Simple Ant Colony Optimization Algorithm

  • Frank Neumann
  • Carsten Witt
Open Access


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.


Randomized search heuristics Ant colony optimization Runtime analysis 


  1. 1.
    Dorigo, M., Blum, C.: Ant colony optimization theory: A survey. Theor. Comput. Sci. 344, 243–278 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT, Cambridge (2004) zbMATHGoogle Scholar
  3. 3.
    Dorigo, M., Maniezzo, V., Colorni, A.: The ant system: An autocatalytic optimizing process. Tech. Rep. 91-016 Revised, Politecnico di Milano, Italy (1991) Google Scholar
  4. 4.
    Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edn., vol. 1. Wiley, New York (1968) zbMATHGoogle Scholar
  6. 6.
    Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd edn., vol. 2. Wiley, New York (1971) zbMATHGoogle Scholar
  7. 7.
    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) Google Scholar
  8. 8.
    Gleser, L.J.: On the distribution of the number of successes in independent trials. Ann. Probab. 3(1), 182–188 (1975) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gutjahr, W.J.: A generalized convergence result for the graph-based ant system metaheuristic. Probab. Eng. Inf. Sci. 17, 545–569 (2003) zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Gutjahr, W.J.: On the finite-time dynamics of ant colony optimization. Methodol. Comput. Appl. Probab. 8, 105–133 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jerrum, M., Sorkin, G.B.: The Metropolis algorithm for graph bisection. Discrete Appl. Math. 82(1–3), 155–175 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    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) Google Scholar
  13. 13.
    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) CrossRefGoogle Scholar
  14. 14.
    Scheideler, C.: Probabilistic methods for coordination problems. HNI-Verlagsschriftenreihe 78. Habilitation Thesis, University of Paderborn. Available at (2000)
  15. 15.
    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) Google Scholar
  16. 16.
    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) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Algorithms and ComplexityMax-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.FB Informatik, LS 2Universität DortmundDortmundGermany

Personalised recommendations