Abstract
Given a complete graph G = (V,E)with a cost function c : E →ℝ + and a vertex subset R ⊂ V, an internal Steiner tree is a Steiner tree which contains all vertices in R such that each vertex in R is restricted to be an internal vertex. The internal Steiner tree problem is to find an internal Steiner tree T whose total costs ∑ (u,v) ∈ E(T) c(u,v) is minimum. In this paper, we first show that the internal Steiner tree problem is MAX SNP-hard. We then present an approximation algorithm with approximation ratio 2ρ + 1 for the problem, where ρ is the best known approximation ratio for the Steiner tree problem.
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
Preview
Unable to display preview. Download preview PDF.
References
Arora, S., et al.: Proof verification and the hardness of approximation problems. Journal of the Association for Computing Machinery 45, 501–555 (1998)
Bern, M., Plassmann, P.: The Steiner problem with edge lengths 1 and 2. Information Processing Letters 32(4), 171–176 (1989)
Cheng, X., Du, D.Z.: Steiner Trees in Industry. Kluwer Academic Publishers, Dordrecht (2001)
Cormen, T.H., et al.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)
Du, D.Z., Smith, J.M., Rubinstein, J.H.: Advance in Steiner Tree. Kluwer Academic Publishers, Dordrecht (2000)
Garey, M., Graham, R., Johnson, D.: The complexity of computing Steiner minimal trees. SIAM Journal on Applied Mathematics 32, 835–859 (1977)
Garey, M., Johnson, D.: The rectilinear Steiner problem is NP-complete. SIAM Journal on Applied Mathematics 32, 826–834 (1977)
Hwang, F.K., Richards, D.S., Winter, P.: The Steiner Tree Problem. Annals of Discrete Mathematics, vol. 53. Elsevier Science Publishers, Amsterdam (1992)
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)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. In: Proceedings of the 20th ACM Symposium on Theory of Computing, pp. 229–234 (1988)
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)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hsieh, SY., Gao, HM., Yang, SC. (2007). On the Internal Steiner Tree Problem. In: Cai, JY., Cooper, S.B., Zhu, H. (eds) Theory and Applications of Models of Computation. TAMC 2007. Lecture Notes in Computer Science, vol 4484. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72504-6_25
Download citation
DOI: https://doi.org/10.1007/978-3-540-72504-6_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72503-9
Online ISBN: 978-3-540-72504-6
eBook Packages: Computer ScienceComputer Science (R0)