Abstract
We present a heuristic for the Euclidean Steiner tree problem in ℜ d for d≥2. The algorithm utilizes the Delaunay triangulation to generate candidate Steiner points for insertion, the minimum spanning tree to identify the Steiner points to remove, and second-order cone programming to optimize the location of the remaining Steiner points. Unlike other ESTP heuristics relying upon Delaunay triangulation, we insert Steiner points probabilistically into Delaunay triangles to achieve different subtrees on subsets of terminal points. We govern this neighbor generation procedure with a local search framework that extends effectively into higher dimensions. We present computational results on benchmark test problems in ℜ d for 2≤d≤5.
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References
Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. Assoc. Comput. Mach. 45(5), 753–782 (1998)
Beasley, J.E.: OR-Library: Distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)
Beasley, J.E., Goffinet, F.: A Delaunay triangulation-based heuristic for the Euclidean Steiner problem. Networks 24(4), 215–224 (1994)
Chung, F.R.K., Graham, R.L.: Steiner trees for ladders. Ann. Discrete Math. 2, 173–200 (1978)
Dreyer, D.R., Overton, M.L.: Two heuristics for the Euclidean Steiner tree problem. J. Glob. Optim. 13(1), 95–106 (1998)
Du, D.Z., Hwang, F.K.: A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica 7(1), 121–135 (1992)
Fampa, M., Anstreicher, K.M.: An improved algorithm for computing Steiner minimal trees in Euclidean d-space. Discrete Optim. 5(2), 530–540 (2008)
Garey, M.R., Graham, R.L., Johnson, D.S.: The complexity of computing Steiner minimal trees. SIAM J. Appl. Math. 32(4), 835–859 (1977)
Gilbert, E.N., Pollak, H.O.: Steiner minimal trees. SIAM J. Appl. Math. 16(1), 1–29 (1968)
Montgomery, D.C.: Design and Analysis of Experiments, 5th edn. Wiley, New York (2001)
Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction, 2nd edn. Springer, New York (1988)
Rao, S.B., Smith, W.D.: Improved approximation schemes for geometrical graphs via “spanners” and “banyans”. In 30th ACM Symposium on Theory of Computing, pp. 540–550 (1998)
Ravada, S., Sherman, A.T.: Experimental evaluation of a partitioning algorithm for the Steiner tree problem in R 2 and R 3. Networks 24(8), 409–415 (1994)
Smith, W.D.: How to find Steiner minimal trees in Euclidean d-space. Algorithmica 7(1), 137–177 (1992)
Smith, J.M., Lee, D.T., Liebman, J.S.: An O(Nlog N) heuristic for Steiner minimal tree problems on the Euclidean metric. Networks 11, 23–29 (1981)
Smith, J.M., Weiss, R., Patel, M.: An O(N 2) heuristic for Steiner minimal trees in E 3. Networks 26(4), 273–289 (1995)
Sturm, J.F., Polik, I.: SeDuMi: self dual minimization version 1.1 (2006)
Thompson, E.A.: The method of minimum evolution, Ann. Hum. Genet. 36(3), 333–340 (1973)
Toppur, B., Smith, J.M.G.: A sausage heuristic for Steiner minimal trees in three-dimensional Euclidean space. J. Math. Model. Algorithms 4(2), 199–217 (2005)
Warme, D.M., Winter, P., Zachariasen, M.: GeoSteiner 3.1. Department of Computer Science, University of Copenhagen (DIKU) (2001)
Zachariasen, M.: Local search for the Steiner tree problem in the Euclidean plane. Eur. J. Oper. Res. 119(2), 282–300 (1999)
Zachariasen, M., Winter, P.: Concatenation-based greedy heuristics for the Euclidean Steiner tree problem. Algorithmica 25(4), 418–437 (1999)
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Van Laarhoven, J.W., Ohlmann, J.W. A randomized Delaunay triangulation heuristic for the Euclidean Steiner tree problem in ℜ d . J Heuristics 17, 353–372 (2011). https://doi.org/10.1007/s10732-010-9137-z
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DOI: https://doi.org/10.1007/s10732-010-9137-z