An Improved Approximation Algorithm for the Terminal Steiner Tree Problem

  • Yen Hung Chen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6784)


Given a complete graph G = (V,E) with a length function on edges and a subset R of V, the terminal Steiner tree is defined to be a Steiner tree in G with all the vertices of R as its leaves. Then the terminal Steiner tree problem is to find a terminal Steiner tree in G with minimum length. In this paper, we present an approximation algorithm with performance ratio \(2\rho-\frac{(\rho\alpha^2-\alpha\rho)}{(\alpha+\alpha^2)(\rho-1)+2(\alpha-1)^2}\) for the terminal Steiner tree problem, where ρ is the best-known performance ratio for the Steiner tree problem with any α ≥ 2. When we let α = 3.87 ≈ 4, this result improves the previous performance ratio of 2.515 to 2.458.


Approximation algorithms NP-complete Steiner tree terminal Steiner tree problem multicast routing evolutionary tree reconstruction in biology telecommunications 


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  1. 1.
    Berman, P., Ramaiyer, V.: Improved Approximations for the Steiner Tree Problem. Journal of Algorithms 17, 381–408 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bern, M., Plassmann, P.: The Steiner Tree Problem with Edge Lengths 1 and 2. Information Processing Letters 32, 171–176 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Borchers, A., Du, D.Z.: The k-Steiner Ratio in Graphs. SIAM Journal on Computing 26, 857–869 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Caldwell, A., Kahng, A., Mantik, S., Markov, I., Zelikovsky, A.: On Wirelength Estimations for Row-Based Placement. In: Proceedings of the 1998 International Symposium on Physical Design (ISPD 1998), pp. 4–11. ACM, Monterey (1998)CrossRefGoogle Scholar
  5. 5.
    Chen, Y.H., Lu, C.L., Tang, C.Y.: On the Full and Bottleneck Full Steiner Tree Problems. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 122–129. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Cheng, X., Du, D.Z.: Steiner Tree in Industry. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  7. 7.
    Drake, D.E., Hougardy, S.: On Approximation Algorithms for the Terminal Steiner Tree Problem. Information Processing Letters 89, 15–18 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Du, D.Z., Smith, J.M., Rubinstein, J.H.: Advances in Steiner Tree. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  9. 9.
    Du, D.Z., Hu, X.: Steiner Tree Problems in Computer Communication Networks. World Scientific Publishing Company, Singapore (2008)CrossRefzbMATHGoogle Scholar
  10. 10.
    Fuchs, B.: A Note on the Terminal Steiner Tree Problem. Information Processing Letters 87, 219–220 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Garey, M.R., Graham, R.L., Johnson, D.S.: The Complexity of Computing Steiner Minimal Trees. SIAM Journal of Applied Mathematics 32, 835–859 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Graur, D., Li, W.H.: Fundamentals of Molecular Evolution, 2nd edn. Sinauer Publishers, Sunderland (2000)Google Scholar
  13. 13.
    Hougardy, S., Prommel, H.J.: A 1.598 Approximation Algorithm for the Steiner Problem in Graphs. In: Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1999), pp. 448–453. ACM/SIGACT-SIAM, Baltimore (1999)Google Scholar
  14. 14.
    Hsieh, S.Y., Gao, H.M.: On the Partial Terminal Steiner Tree Problem. The Journal of Supercomputing 41, 41–52 (2007)CrossRefGoogle Scholar
  15. 15.
    Hsieh, S.Y., Yang, S.C.: Approximating the Selected-Internal Steiner Tree. Theoretical Computer Science 381, 288–291 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Hwang, F.K., Richards, D.S., Winter, P.: The Steiner Tree Problem. Annuals of Discrete Mathematics, vol. 53. North-Holland, Elsevier, Amsterdam (1992)zbMATHGoogle Scholar
  17. 17.
    Kahng, A.B., Robins, G.: On Optimal Interconnections for VLSI. Kluwer Academic Publishers, Boston (1995)CrossRefzbMATHGoogle Scholar
  18. 18.
    Karpinski, M., Zelikovsky, A.: New Approximation Algorithms for the Steiner Tree Problems. Journal of Combinatorial Optimization 1, 47–65 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Kim, J., Warnow, T.: Tutorial on Phylogenetic Tree Estimation. Department of Ecology and Evolutionary Biology. Yale University, New Haven (1999) (manuscript)Google Scholar
  20. 20.
    Lin, G.H., Xue, G.L.: On the Terminal Steiner Tree Problem. Information Processing Letters 84, 103–107 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Lu, C.L., Tang, C.Y., Lee, R.C.T.: The Full Steiner Tree Problem. Theoretical Computer Science 306, 55–67 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Martineza, F.V., Pinab, J.C.D., Soares, J.: Algorithm for Terminal Steiner Trees. Theoretical Computer Science 389, 133–142 (2007)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Prommel, H.J., Steger, A.: A New Approximation Algorithm for the Steiner Tree Problem with Performance Ratio 5/3. Journal of Algorithms 36, 89–101 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Robins, G., Zelikovsky, A.: Improved Steiner Tree Approximation in Graphs. In: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2000), pp. 770–779. ACM/SIGACT-SIAM, San Francisco (2000)Google Scholar
  25. 25.
    Robins, G., Zelikovsky, A.: Tighter Bounds for Graph Steiner Tree Approximation. SIAM Journal on Discrete Mathematics 19, 122–134 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Zelikovsky, A.: An 11/6-Approximation Algorithm for the Network Steiner Problem. Algorithmica 9, 463–470 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Zelikovsky, A.: A Faster Approximation Algorithm for the Steiner Tree Problem in Graphs. Information Processing Letters 46, 79–83 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Zelikovsky, A.: Better Approximation Bounds for the Network and Euclidean Steiner Tree Problems. Technical report CS-96-06, University of Virginia (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Yen Hung Chen
    • 1
  1. 1.Department of Computer ScienceTaipei Municipal University of EducationTaipeiTaiwan, R.O.C.

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