Abstract
A rigorous runtime analysis of evolutionary multi-objective optimization for the classical vertex cover problem in the context of parameterized complexity analysis has been presented by Kratsch and Neumann [1]. In this paper, we extend the analysis to the weighted vertex cover problem and provide a fixed parameter evolutionary algorithm with respect to OPT, the cost of the optimal solution for the problem. Moreover, using a diversity mechanism, we present a multi-objective evolutionary algorithm that finds a \(2-\)approximation in expected polynomial time.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Kratsch, S., Neumann, F.: Fixed-parameter evolutionary algorithms and the vertex cover problem. Algorithmica 65(4), 754–771 (2013)
Neumann, F., Witt, C.: Bioinspired Computation in Combinatorial Optimization: Algorithms and Their Computational Complexity, 1st edn. Springer, New York (2010)
Auger, A., Doerr, B.: Theory of Randomized Search Heuristics: Foundations and Recent Developments. World Scientific Publishing Co., Inc., River Edge (2011)
Jansen, T.: Analyzing Evolutionary Algorithms - The Computer Science Perspective. NCS. Springer, Berlin (2013)
Sutton, A.M., Neumann, F.: A parameterized runtime analysis of simple evolutionary algorithms for makespan scheduling. In: Coello, C.A.C., Cutello, V., Deb, K., Forrest, S., Nicosia, G., Pavone, M. (eds.) PPSN 2012, Part I. LNCS, vol. 7491, pp. 52–61. Springer, Heidelberg (2012)
Sutton, A.M., Neumann, F., Nallaperuma, S.: Parameterized runtime analyses of evolutionary algorithms for the planar euclidean traveling salesperson problem. Evol. Comput. 22(4), 595–628 (2014)
Friedrich, T., Hebbinghaus, N., Neumann, F., He, J., Witt, C.: Approximating covering problems by randomized search heuristics using multi-objective models. In: Proceedings of 9th Annual Conference on Genetic and Evolutionary Computation, GECCO 2007, pp. 797–804. ACM, New York (2007)
Friedrich, T., He, J., Hebbinghaus, N., Neumann, F., Witt, C.: Analyses of simple hybrid algorithms for the vertex cover problem. Evol. Comput. 17(1), 3–19 (2009)
Oliveto, P.S., He, J., Yao, X.: Analysis of the (1+1)-EA for finding approximate solutions to vertex cover problems. IEEE Trans. Evol. Comput. 13(5), 1006–1029 (2009)
Jansen, T., Oliveto, P.S., Zarges, C.: Approximating vertex cover using edge-based representations. In: Neumann, F., Jong, K.A.D. (eds.) Foundations of Genetic Algorithms XII, FOGA 2013, Adelaide, SA, Australia, 16–20 January 2013, pp. 87–96. ACM (2013)
Pourhassan, M., Gao, W., Neumann, F.: Maintaining 2-approximations for the dynamic vertex cover problem using evolutionary algorithms. In: Proceedings of Genetic and Evolutionary Computation Conference, GECCO 2015, Madrid, Spain, pp. 903–910. ACM (2015)
Laumanns, M., Thiele, L., Deb, K., Zitzler, E.: Combining convergence and diversity in evolutionary multiobjective optimization. Evol. Comput. 10(3), 263–282 (2002)
Horoba, C., Neumann, F.: Benefits and drawbacks for the use of \(\epsilon \)-dominance in evolutionary multi-objective optimization. In: Proceedings of GECCO 2008 (2008)
Neumann, F., Reichel, J., Skutella, M.: Computing minimum cuts by randomized search heuristics. Algorithmica 59(3), 323–342 (2011)
Neumann, F., Reichel, J.: Approximating minimum multicuts by evolutionary multi-objective algorithms. In: Rudolph, G., Jansen, T., Lucas, S., Poloni, C., Beume, N. (eds.) PPSN 2008. LNCS, vol. 5199, pp. 72–81. Springer, Heidelberg (2008)
Balinski, M.: On the maximum matching, minimum covering. In: Proceedings of Symposium on Mathematical Programming, pp. 434–445. Princeton University Press (1970)
Pourhassan, M., Shi, F., Neumann, F.: Parameterized analysis of multi-objective evolutionary algorithms and the weighted vertex cover problem (2016). CoRR http://arXiv.org/abs/1604.01495
Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64(4), 673–697 (2012)
Acknowledgements
This research has been supported by Australian Research Council grants DP140103400 and DP160102401.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Pourhassan, M., Shi, F., Neumann, F. (2016). Parameterized Analysis of Multi-objective Evolutionary Algorithms and the Weighted Vertex Cover Problem. In: Handl, J., Hart, E., Lewis, P., López-Ibáñez, M., Ochoa, G., Paechter, B. (eds) Parallel Problem Solving from Nature – PPSN XIV. PPSN 2016. Lecture Notes in Computer Science(), vol 9921. Springer, Cham. https://doi.org/10.1007/978-3-319-45823-6_68
Download citation
DOI: https://doi.org/10.1007/978-3-319-45823-6_68
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45822-9
Online ISBN: 978-3-319-45823-6
eBook Packages: Computer ScienceComputer Science (R0)