Abstract
Given a fixed set of points in a N-dimensional space (N ≥ 3) with Euclidean metrics, the Euclidean Steiner Tree Problem in R N consists of finding a minimum length tree that spans all these points using, if necessary, extra points (Steiner points). The finding of such solution is a NP-hard problem. This paper presents a modified metaheuristic based on Improved Particle Swarm Optimization to the problem considered. Finally, computational experiments compare the performance of the proposed heuristic, considering solution’s quality and computational time, in regard to previous works in the literature.
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References
Alford, C., Brazil, M., Lee, D.H.: Optimisation in Underground Mining, pp. 561–577. Springer, Boston (2007). https://doi.org/10.1007/978-0-387-71815-6_30
Brazil, M., Graham, R.L., Thomas, D.A., Zachariasen, M.: On the history of the Euclidean Steiner tree problem. Arch. Hist. Exact Sci. 68(3), 327–354 (2014). https://doi.org/10.1007/s00407-013-0127-z
Du, D., Hu, X.: Steiner Tree Problems in Computer Communication Networks. World Scientific Publishing, River Edge (2008)
Eberhart, R.C., Shi, Y.: Comparing inertia weights and constriction factors in particle swarm optimization. In: Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512), vol. 1, pp. 84–88 (2000). https://doi.org/10.1109/CEC.2000.870279
Fampa, M., Anstreicher, K.M.: An improved algorithm for computing Steiner minimal trees in Euclidean d-space. Discret. Optim. 5(2), 530–540 (2008). https://doi.org/10.1016/j.disopt.2007.08.006
Garey, M.R., Johnson, D.S.: Computers and Intractability; a Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979)
Garey, M.R., Graham, R.L., Johnson, D.S.: The complexity of computing Steiner minimal trees. SIAM J. Appl. Math. 32 (1977). https://doi.org/10.1137/0132072
Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948 (1995). https://doi.org/10.1109/ICNN.1995.488968
Kuhn, H.W.: “Steiner’s” Problem Revisited, pp. 52–70. Mathematical Association of America, Washington (1974)
Montenegro, F., Torreão, J.R.A., Maculan, N.: Microcanonical optimization algorithm for the Euclidean Steiner problem in Rn with application to phylogenetic inference. Phys. Rev. E 68 (2003). https://doi.org/10.1103/PhysRevE.68.056702
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1988)
Prim, R.C.: Shortest connection networks and some generalizations. Bell Syst. Tech. J. 36(6), 1389–1401 (1957). https://doi.org/10.1002/j.1538-7305.1957.tb01515.x
Rocha, M.L.: An hybrid metaheuristic approach to solve the Euclidean Steiner tree problem in Rn. In: Proceedings of XLV Brazilian Symposium on Operational Research, vol. 1, pp. 1881–1892 (2013)
Smith, W.D.: How to find Steiner minimal trees in Euclidean d-space. Algorithmica 7(1), 137–177 (1992). https://doi.org/10.1007/BF01758756
van den Bergh, F., Engelbrecht, A.: A study of particle swarm optimization particle trajectories. Inf. Sci. 176(8), 937–971 (2006). https://doi.org/10.1016/j.ins.2005.02.003
Zachariasen, M.: Local search for the Steiner tree problem in the Euclidean plane. Eur. J. Oper. Res. 119(2), 282–300 (1999). https://doi.org/10.1016/S0377-2217(99)00131-9
Acknowledgements
The authors acknowledge the reviewers for important and helpful contributions to this work. The development of this research benefited from the UFT Institutional Productivity Research Program (PROPESQ/UFT).
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Costa, W.W., Rocha, M.L., Prata, D.N., Moreira, P.L. (2019). Application of Enhanced Particle Swarm Optimization in Euclidean Steiner Tree Problem Solving in R N . In: Platt, G., Yang, XS., Silva Neto, A. (eds) Computational Intelligence, Optimization and Inverse Problems with Applications in Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-96433-1_4
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