Abstract
In the Steiner Tree Problem in Graphs, a subset of nodes, called terminals, must be efficiently interconnected. The graph is undirected and weighted, and candidate solutions can include additional nodes called Steiner vertices. The problem is NP-complete, and optimization algorithms based on population metaheuristics, e.g., Genetic Algorithms (GAs), have been proposed. However, traditional recombination operators may produce inefficient solutions for graph-based problems. We propose a new recombination operator for the Steiner Tree Problem based on the graph representation. The new operator is a kind of partition crossover: it breaks the graph formed by the union of two parents solutions and decomposes the evaluation function by finding connected subgraphs. We also investigate two soft-repair operators that produce small changes in candidate solutions: an MST transformation and a pruning repair. The GA with the proposed crossover operator was able to find the global optimum solution for all tested instances and has a success rate of 28% in the worst case. The experiments with the proposed crossover presented a quick convergence to optimal solutions, and an average solution cost lower in most cases than other approaches.
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
Beasley, J.E.: OR-library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Algoritmos: Teoria e Prática. Editora Elsevier (2012)
Esbensen, H.: Computing near-optimal solutions to the Steiner problem in a graph using a genetic algorithm. Networks 26(4), 173–185 (1995). https://doi.org/10.1002/net.3230260403
Hakimi, S.L.: Steiner’s problem in graphs and its implications. Networks 1(2), 113–133 (1971). https://doi.org/10.1002/net.3230010203
Kapsalis, A., Rayward-smith, V.J., Smith, G.D.: Solving the graphical Steiner tree problem using genetic algorithms. J. Oper. Res. Soc. 44(4), 397–406 (1993)
Karp, R.M.: Reducibility among Combinatorial Problems, pp. 85–103. Springer, Boston (1972). https://doi.org/10.1007/978-1-4684-2001-2_9
Kou, L., Markowsky, G., Berman, L.: A fast algorithm for Steiner trees. Acta Inf. 15(2), 141–145 (1981). https://doi.org/10.1007/BF00288961
Kruskal, J.B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 7(1), 48–50 (1956). http://www.jstor.org/stable/2033241
Ljubić, I.: Solving Steiner trees: recent advances, challenges, and perspectives. Networks 77(2), 177–204 (2021)
Prim, R.C.: Shortest connection networks and some generalizations. Bell Syst. Tech. J. 36(6), 1389–1401 (1957)
Raidl, G.R., Julstrom, B.A.: Edge sets: an effective evolutionary coding of spanning trees. IEEE Trans. Evol. Comput. 7(3), 225–239 (2003). https://doi.org/10.1109/TEVC.2002.807275
Tinós, R., Whitley, D., Ochoa, G.: Generalized Asymmetric Partition Crossover (GAPX) for the asymmetric TSP. In: Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation, GECCO 2014, pp. 501–508. ACM, New York (2014)
Tinós, R., Whitley, D., Ochoa, G.: A new generalized partition crossover for the traveling salesman problem: tunneling between local optima. Evol. Comput. 28(2), 255–288 (2020). https://doi.org/10.1162/evco_a_00254, pMID: 30900928
Whitley, D.: Next generation genetic algorithms: a user’s guide and tutorial. In: Gendreau, M., Potvin, J.-Y. (eds.) Handbook of Metaheuristics. ISORMS, vol. 272, pp. 245–274. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-91086-4_8
Whitley, D., Hains, D., Howe, A.: Tunneling between optima: partition crossover for the traveling salesman problem. In: Proceedings of the 11th Annual Conference on Genetic and Evolutionary Computation, GECCO 2009, pp. 915–922. ACM, New York (2009)
Whitley, D., Hains, D., Howe, A.: A hybrid genetic algorithm for the traveling salesman problem using generalized partition crossover. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN 2010. LNCS, vol. 6238, pp. 566–575. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15844-5_57
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
de Godoi, G.A., Tinós, R., Sanches, D.S. (2021). A Graph-Based Crossover and Soft-Repair Operators for the Steiner Tree Problem. In: Britto, A., Valdivia Delgado, K. (eds) Intelligent Systems. BRACIS 2021. Lecture Notes in Computer Science(), vol 13073. Springer, Cham. https://doi.org/10.1007/978-3-030-91702-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-91702-9_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-91701-2
Online ISBN: 978-3-030-91702-9
eBook Packages: Computer ScienceComputer Science (R0)