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Steiner Tree in Planar Graphs: An O(nlogn) Approximation Scheme with Singly-Exponential Dependence on Epsilon

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Algorithms and Data Structures (WADS 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4619))

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Abstract

We give an algorithm that, for any ε> 0, any undirected planar graph G, and any set S of nodes of G, computes a (1 + ε)-optimal Steiner tree in G that spans the nodes of S. The algorithm takes time O(2poly(1/ε) n logn).

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References

  1. Arora, S.: Polynomial-time approximation schemes for euclidean TSP and other geometric problems. JACM 45(5), 753–782 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Bern, M., Bienstock, D.: Polynomially solvable special cases of the Steiner problem in planar networks. Annals of Operations Research 33, 405–418 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bern, M., Plassmann, P.: The Steiner problem with edge lengths 1 and 2. IPL 32, 171–176 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Borradaile, G., Kenyon-Mathieu, C., Klein, P.: A polynomial-time approximation scheme for Steiner tree in planar graphs. In: 18th SODA, pp. 1285–1294 (2007)

    Google Scholar 

  7. Erickson, R., Monma, C., Veinott, A.: Send-and-split method for minimum-concave-cost network flows. Mathematics of Operations Research 12, 634–664 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  8. Garey, M., Johnson, D.: The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  9. Henzinger, M., Klein, P., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. J. Comput. System Sci. 55(1), 3–23 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hougardy, S., Prömel, H.J.: A 1.598 approximation algorithm for the Steiner problem in graphs. In: 10th SODA, pp. 448–453 (1999)

    Google Scholar 

  11. Karp, R.: On the computational complexity of combinatorial problems. Networks 5, 45–68 (1975)

    MATH  MathSciNet  Google Scholar 

  12. Karpinski, M., Zelikovsky, A.: New approximation algorithms for the Steiner tree problem. Journal of Combinatorial Optimization 1, 47–65 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Klein, P.: A linear-time approximation scheme for planar weighted TSP. In: 46th FOCS, p. 647 (2005)

    Google Scholar 

  14. Kou, L., Markowsky, G., Berman, L.: A fast algorithm for Steiner trees. Acta Informatica 15, 141–145 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mehlhorn, K.: Approximation algorithm for the Steiner problem in graphs. IPL 27(3), 125–128 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mitchell, J.: Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric tsp, k-mst, and related problems. SIAM J. Comput. 28(4), 1298–1309 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Prömel, H.J., Steger, A.: RNC approximation algorithms for the Steiner problem. In: 39th STOC. LNCS, vol. 1200, pp. 559–570. Springer, Heidelberg (1997)

    Google Scholar 

  18. Provan, J.: An approximation scheme for finding Steiner trees with obstacles. SIAM J. Comput. 17, 920–934 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  19. Provan, J.: Convexity and the Steiner tree problem. Networks 18, 55–72 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rao, S., Smith, W.: Approximating geometrical graphs via spanners and banyans. In: 30th STOC, pp. 540–550 (1998)

    Google Scholar 

  21. Robins, G., Zelikovsky, A.: Tighter bounds for graph Steiner tree approximation. SIAM J. Discret. Math. 19(1), 122–134 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Takahashi, H., Matsuyama, A.: An approximate solution for the Steiner problem in graphs. Mathematica Japonicae 24, 571–577 (1980)

    MathSciNet  Google Scholar 

  23. Widmayer, P.: A fast approximation algorithm for Steiner’s problem in graphs. In: Tinhofer, G., Schmidt, G. (eds.) WG 1986. LNCS, vol. 246, pp. 17–28. Springer, Heidelberg (1987)

    Google Scholar 

  24. Wu, Y., Widmayer, P., Wong, C.: A faster approximation algorithm for the Steiner problem in graphs. Acta informatica 23(2), 223–229 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  25. Zelikovsky, A.: Better approximation bounds for the network and Euclidean Steiner tree problems. Technical Report CS-96-06, University of Virginia (1994)

    Google Scholar 

  26. Zelikovsky, A.: An 11/6-approximation algorithm for the network Steiner problem. Algorithmica 9, 463–470 (1999)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Brown University, Providence RI 02912, USA

    Glencora Borradaile, Philip N. Klein & Claire Mathieu

Authors
  1. Glencora Borradaile
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  2. Philip N. Klein
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  3. Claire Mathieu
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Editor information

Frank Dehne Jörg-Rüdiger Sack Norbert Zeh

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© 2007 Springer-Verlag Berlin Heidelberg

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Borradaile, G., Klein, P.N., Mathieu, C. (2007). Steiner Tree in Planar Graphs: An O(nlogn) Approximation Scheme with Singly-Exponential Dependence on Epsilon. In: Dehne, F., Sack, JR., Zeh, N. (eds) Algorithms and Data Structures. WADS 2007. Lecture Notes in Computer Science, vol 4619. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73951-7_25

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  • DOI: https://doi.org/10.1007/978-3-540-73951-7_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73948-7

  • Online ISBN: 978-3-540-73951-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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