MAX-SNP Hardness and Approximation of Selected-Internal Steiner Trees

  • Sun-Yuan Hsieh
  • Shih-Cheng Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4112)


In this paper, we consider an interesting variant of the well-known Steiner tree problem: Given a complete graph G = (V,E) with a cost function c:ER  +  and two subsets R and R′ satisfying R′ ⊂ R ⊆ V, a selected-internal Steiner tree is a Steiner tree which contains (or spans) all the vertices in R such that each vertex in R′ cannot be a leaf. The selected-internal Steiner tree problem is to find a selected-internal Steiner tree with the minimum cost. In this paper, we show that the problem is MAX SNP-hard even when the costs of all edges in the input graph are restricted to either 1 or 2. We also present an approximation algorithm for the problem.


Cost Function Approximation Algorithm Complete Graph Steiner Tree Internal Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the Association for Computing Machinery 45, 501–555 (1998)MATHMathSciNetGoogle Scholar
  2. 2.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamelai, A., Protasi, M.: Complexity and Approximation-Combinatorial Optimization Problems and Their Approximability Properties. Springer, Berlin (1999)MATHGoogle Scholar
  3. 3.
    Bern, M., Plassmann, P.: The Steiner problem with edge lengths 1 and 2. Information Processing Letters 32(4), 171–176 (1989); 5Zelikovsky, On wirelength estimations for row-based placementMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cheng, X., Du, D.Z.: Steiner Trees in Industry. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  5. 5.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)MATHGoogle Scholar
  6. 6.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972); 5 Information Processing Letters, vol. 84(2), pp. 103–107Google Scholar
  7. 7.
    Papadimitriou, C., Yannakakis, M.: Optimization, approximation and complexity classes. Journal of Computer and System Science 43, 770–779 (1991)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Robins, G., Zelikovsky, A.: Improved Steiner tree approximation in graphs. In: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 770–779 (2000)Google Scholar
  9. 9.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sun-Yuan Hsieh
    • 1
  • Shih-Cheng Yang
    • 1
  1. 1.Department of Computer Science and Information EngineeringNational Cheng Kung UniversityTainanTaiwan

Personalised recommendations