Abstract
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.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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). https://doi.org/10.1023/B:JMMA.0000049379.14872.f5
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). https://doi.org/10.1162/EVCO_a_00047
Neumann, F., Witt, C.: Runtime analysis of a simple ant colony optimization algorithm. Algorithmica 54(2), 243–255 (2007). https://doi.org/10.1007/s00453-007-9134-2
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). https://doi.org/10.1145/1276958.1276964
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). https://doi.org/10.1109/MHS.1995.494215
Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948 (1995). https://doi.org/10.1109/ICNN.1995.488968
Schmitt, M., Wanka, R.: Particle swarm optimization almost surely finds local optima. Theoret. Comput. Sci. 561A, 57–72 (2015). https://doi.org/10.1016/j.tcs.2014.05.017
Sudholt, D., Witt, C.: Runtime analysis of a binary particle swarm optimizer. Theoret. Comput. Sci. 411(21), 2084–2100 (2010). https://doi.org/10.1016/j.tcs.2010.03.002
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). https://doi.org/10.1007/978-3-540-39930-8_8
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). https://doi.org/10.1007/978-3-642-23857-4_40
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). https://doi.org/10.1145/3040718.3040721
Sudholt, D., Thyssen, C.: Running time analysis of ant colony optimization for shortest path problems. J. Discrete Algorithms 10, 165–180 (2012). https://doi.org/10.1016/j.jda.2011.06.002
Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, McGraw-Hill, Cambridge (1990)
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). https://doi.org/10.1145/1527125.1527134
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). https://doi.org/10.1007/0-306-48041-7_14
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). https://doi.org/10.1016/B978-155860734-7/50098-6
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). https://arxiv.org/abs/1902.01810, extended version of [11]
Gillespie, J.H.: Some properties of finite populations experiencing strong selection and weak mutation. Am. Nat. 121(5), 691–708 (1983). https://doi.org/10.1086/284095
Acknowledgement
The authors would like to thank Bernd Bassimir for useful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Raß, A., Schreiner, J., Wanka, R. (2019). Runtime Analysis of Discrete Particle Swarm Optimization Applied to Shortest Paths Computation. In: Liefooghe, A., Paquete, L. (eds) Evolutionary Computation in Combinatorial Optimization. EvoCOP 2019. Lecture Notes in Computer Science(), vol 11452. Springer, Cham. https://doi.org/10.1007/978-3-030-16711-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-16711-0_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-16710-3
Online ISBN: 978-3-030-16711-0
eBook Packages: Computer ScienceComputer Science (R0)