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On the partial terminal Steiner tree problem

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Abstract

We investigate a practical variant of the well-known graph Steiner tree problem. For a complete graph G = ( V,E ) with length function l:E → R + and two vertex subsets R ⊂ V and R ′ ⊆ R, a partial terminal Steiner tree is a Steiner tree which contains all vertices in R such that all vertices in R ∖ R ′ belong to the leaves of this Steiner tree. The partial terminal Steiner tree problem is to find a partial terminal Steiner tree T whose total lengths ∑ (u,v) ∈T l ( u,v ) is minimum. In this paper, we show that the problem is both NP-complete and MAX SNP-hard when the lengths of edges are restricted to either 1 or 2. We also provide an approximation algorithm for the problem.

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References

  1. Arora S, Lund C, Motwani R, Sudan M, Szegedy M (1998) Proof verification and the hardness of approximation problems. J Assoc Comput Mach 45:501–555

    MATH  MathSciNet  Google Scholar 

  2. Berman P, Ramaiyer V (1994) Improved approximations for the Steiner tree problem. J Algorithms 17(3):381–408

    Article  MATH  MathSciNet  Google Scholar 

  3. Bern M (1990) Faster exact algorithms for Steiner tree in planar networks. Networks 20:109–120

    MATH  MathSciNet  Google Scholar 

  4. Bern M, Plassmann P (1989) The Steiner problem with edge lengths 1 and 2. Inf Process Lett 32(4):171–176

    Article  MATH  MathSciNet  Google Scholar 

  5. Borchers A, Du DZ (1997) The k-Steiner ratio in graphs. SIAM J Comput 26(3):857–869

    Article  MATH  MathSciNet  Google Scholar 

  6. Caldewll A, Kahng A, Mantik S, Markov I, Zelikovsky A (1998) On wirelength estimations for row-based placement. In: Proceedings of the 1998 international symposium on physical design (ISPD), pp 4–11

  7. Chen YH, Lu CL, Tang CY (2003) On the full and bottleneck full Steiner tree problems. In: Proceedings of the 9th annual international conference on computing and combinatorics (COCOON), LNCS vol 2697, pp 122–129

  8. Cheng X, Du DZ (2001) Steiner trees in industry. Kluwer Academic, Dordrecht

    Google Scholar 

  9. Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to algorithms, 2nd edn. MIT Press, Cambridge

    Google Scholar 

  10. Drake DE, Hougardy S (2004) On approximation algorithms for the terminal Steiner tree problem. Inf Process Lett 89(1):15–18

    Article  MathSciNet  Google Scholar 

  11. Du DZ (1995) On component-size bounded Steiner trees. Discret Appl Math 60:131–140

    Article  MATH  Google Scholar 

  12. Du DZ, Smith JM, Rubinstein JH (2000) Advance in Steiner tree. Kluwer Academic, Dordrecht

    Google Scholar 

  13. Foulds L, Graham R (1982) The Steiner problem in phylogeny is NP-complete. Adv Appl Math 3:43–49

    Article  MATH  MathSciNet  Google Scholar 

  14. Garey M, Graham R, Johnson D (1977) The complexity of computing Steiner minimal trees. SIAM J Appl Math 32(4):835–859

    Article  MATH  MathSciNet  Google Scholar 

  15. Garey M, Johnson D (1977) The rectilinear Steiner problem is NP-complete. SIAM J Appl Math 32:826–834

    Article  MATH  MathSciNet  Google Scholar 

  16. Garg N, Konjevod G, Ravi R (2000) A polylogarithmic approximation algorithm for the group Steiner tree problem. J Algorithm 37(1):66–84

    Article  MATH  MathSciNet  Google Scholar 

  17. Graur D, Li WH (2000) Fundamentals of molecular evolution, 2nd edn. Sinauer, Sunderland

    Google Scholar 

  18. Halperin E, Krauthgamer R (2003) Polylogarithmic inapproximability. In: Proceedings of the 35th ACM symposium on theory of computing, pp 585–594

  19. Hougardy S, Prömel HJ (1999) A 1.598 approximation algorithm for the Steiner tree problem in graphs. In: Proceedings of the 10th annual ACM-SIAM symposium on discrete algorithms (SODA), pp 448–453

  20. Hwang FK, Richards DS, Winter P (1992) The Steiner tree problem. In: Annuals of discrete mathematices 53. Elsevier Science, Amsterdam

    Google Scholar 

  21. Kahng AB, Robins G (1995) On optimal interconnections for VLSI. Kluwer Academic

  22. Karp R (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations. Plenum Press, New York, pp 85–103

  23. Kim J, Warnow T (1999) Tutorial on phylogenetic tree estimation, manuscript, Department of Ecology and Evolutionary Biology, Yale University

  24. Lin GH, Xue GL (2002) On the terminal Steiner tree problem. Inf Process Lett 84(2):103–107

    Article  MATH  MathSciNet  Google Scholar 

  25. Lu CL, Tang CY, Lee RCT (2003) The full Steiner tree problem in phylogeny. Theor Comput Sci 306(1–3):55–67

    Article  MATH  MathSciNet  Google Scholar 

  26. Papadimitriou CH, Yannakakis M (1988) Optimization, approximation, and complexity classes. In: Proceedings of the 20th ACM symposium on theory of computing, pp 229–234

  27. Robins G, Zelikovsky A (2005) Improved Steiner tree approximation in graphs. SIAM J Discret Math 19(1):122–134

    Article  MATH  MathSciNet  Google Scholar 

  28. West DB (2001) Introduction to graph theory, 2nd edn. Prentice Hall, Upper Saddle River

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Information Engineering, National Cheng Kung University, No 1. University Road, Tainan, 701, Taiwan

    Sun-Yuan Hsieh & Huang-Ming Gao

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  1. Sun-Yuan Hsieh
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  2. Huang-Ming Gao
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Correspondence to Sun-Yuan Hsieh.

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Hsieh, SY., Gao, HM. On the partial terminal Steiner tree problem. J Supercomput 41, 41–52 (2007). https://doi.org/10.1007/s11227-007-0102-z

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