Runtime Analysis of Discrete Particle Swarm Optimization Applied to Shortest Paths Computation

  • Alexander RaßEmail author
  • Jonas Schreiner
  • Rolf Wanka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11452)


We mathematically analyze a discrete particle swarm optimization (PSO) algorithm solving the single-source shortest path (SSSP) problem. Key features are an improved and extended study on Markov chains expanding the adaptability of this technique and its application on the well-known SSSP problem. The results are upper and lower bounds on the expected optimization time. For upper bounds, we combine return times within a Markov model with the well known fitness level method which is appropriate even for the non-elitist PSO algorithm. For lower bounds we prove that the recently introduced property of indistinguishability applies in this setting and we also combine it with a further Markov chain analysis. We prove a cubic upper and a quadratic lower bound and an exponential upper and lower bound on the expected runtime, respectively, depending on a PSO parameter.


Discrete particle swarm optimization Runtime analysis Single-source shortest paths Markov chains 



The authors would like to thank Bernd Bassimir for useful discussions.


  1. 1.
    Scharnow, J., Tinnefeld, K., Wegener, I.: The analysis of evolutionary algorithms on sorting and shortest paths problems. J. Math. Model. Algorithms 3(4), 349–366 (2004). Scholar
  2. 2.
    Doerr, B., Happ, E., Klein, C.: Tight analysis of the (1+1)-EA for the single source shortest path problem. Evol. Comput. 19(4), 673–691 (2011). Scholar
  3. 3.
    Neumann, F., Witt, C.: Runtime analysis of a simple ant colony optimization algorithm. Algorithmica 54(2), 243–255 (2007). Scholar
  4. 4.
    Doerr, B., Neumann, F., Sudholt, D., Witt, C.: On the runtime analysis of the 1-ANT ACO algorithm. In: Proceedings of the 9th ACM Genetic and Evolutionary Computation Conference (GECCO), pp. 33–40 (2007).
  5. 5.
    Eberhart, R.C., Kennedy, J.: A new optimizer using particle swarm theory. In: Proceedings of the 6th International Symposium on Micro Machine and Human Science, pp. 39–43 (1995).
  6. 6.
    Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948 (1995).
  7. 7.
    Schmitt, M., Wanka, R.: Particle swarm optimization almost surely finds local optima. Theoret. Comput. Sci. 561A, 57–72 (2015). Scholar
  8. 8.
    Sudholt, D., Witt, C.: Runtime analysis of a binary particle swarm optimizer. Theoret. Comput. Sci. 411(21), 2084–2100 (2010). Scholar
  9. 9.
    Clerc, M.: Discrete particle swarm optimization, illustrated by the traveling salesman problem. In: New Optimization Techniques in Engineering, vol. 141, pp. 219–239. Springer, Heidelberg (2004).
  10. 10.
    Hoffmann, M., Mühlenthaler, M., Helwig, S., Wanka, R.: Discrete particle swarm optimization for TSP: theoretical results and experimental evaluations. In: Bouchachia, A. (ed.) ICAIS 2011. LNCS (LNAI), vol. 6943, pp. 416–427. Springer, Heidelberg (2011). Scholar
  11. 11.
    Mühlenthaler, M., Raß, A., Schmitt, M., Siegling, A., Wanka, R.: Runtime analysis of a discrete particle swarm optimization algorithm on sorting and OneMax. In: Proceedings of the 14th ACM/SIGEVO Workshop on Foundations of Genetic Algorithms (FOGA), pp. 13–24 (2017).
  12. 12.
    Sudholt, D., Thyssen, C.: Running time analysis of ant colony optimization for shortest path problems. J. Discrete Algorithms 10, 165–180 (2012). Scholar
  13. 13.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, McGraw-Hill, Cambridge (1990)zbMATHGoogle Scholar
  14. 14.
    Baswana, S., Biswas, S., Doerr, B., Friedrich, T., Kurur, P.P., Neumann, F.: Computing single source shortest paths using single-objective fitness. In: Proceedings of the 10th ACM/SIGEVO Workshop on Foundations of Genetic Algorithms (FOGA), pp. 59–66 (2009).
  15. 15.
    Wegener, I.: Methods for the analysis of evolutionary algorithms on pseudo-Boolean functions. In: Sarker, R., et al. (eds.) Evolutionary Optimization, pp. 349–369. Springer, Boston (2002). Scholar
  16. 16.
    Droste, S., Jansen, T., Wegener, I.: Dynamic parameter control in simple evolutionary algorithms. In: Proceedings of the 6th ACM/SIGEVO Workshop on Foundations of Genetic Algorithms (FOGA), pp. 275–294 (2001).
  17. 17.
    Mühlenthaler, M., Raß, A., Schmitt, M., Wanka, R.: Exact Markov chain-based runtime analysis of a discrete particle swarm optimization algorithm on sorting and OneMax (2019)., extended version of [11]
  18. 18.
    Gillespie, J.H.: Some properties of finite populations experiencing strong selection and weak mutation. Am. Nat. 121(5), 691–708 (1983). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Erlangen-NurembergErlangenGermany

Personalised recommendations