Skip to main content
SpringerLink
Log in

Primal-Dual Approaches to the Steiner Problem

  • Conference paper
  • First Online:
Approximation Algorithms for Combinatorial Optimization (APPROX 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1913))

Abstract

We study several old and new algorithms for computing lower and upper bounds for the Steiner problem in networks using dualascent and primal-dual strategies. We show that none of the known algorithms can both generate tight lower bounds empirically and guarantee their quality theoretically; and we present a new algorithm which combines both features. The new algorithm has running time O(re logn) and guarantees a ratio of at most two between the generated upper and lower bounds, whereas the fastest previous algorithm with comparably tight empirical bounds has running time O(e 2) without a constant approximation ratio. Furthermore, we show that the approximation ratio two between the bounds can even be achieved in time O(e + n log n), improving the previous time bound of O(n 2 log n).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Y. P. Aneja. An integer linear programming approach to the Steiner problem in graphs. Networks, 10:167–178, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Chopra, E. R. Gorres, and M. R. Rao. Solving the Steiner tree problem on a graph using branch and cut. ORSA Journal on Computing, 4:320–335, 1992.

    MATH  Google Scholar 

  3. S. Chopra and M. R. Rao. The Steiner tree problem I: Formulations, compositions and extension of facets. Mathematical Programming, pages 209–229, 1994.

    Google Scholar 

  4. T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms. MIT Press, 1990.

    Google Scholar 

  5. C. W. Duin. Steiner’s Problem in Graphs. PhD thesis, Amsterdam University, 1993.

    Google Scholar 

  6. H. N. Gabow, M. X. Goemans, and D. P. Williamson. An efficient approximation algorithm for the survivable network design problem. In Proceedings 3rd Symposium on Integer Programming and Combinatorial Opt., pages 57–74, 1993.

    Google Scholar 

  7. M. X. Goemans. Personal communication, 1998.

    Google Scholar 

  8. M. X. Goemans and D. J. Bertsimas. Survivable networks, linear programming relaxations and the parsimonious property. Mathematical Programming, 60:145–166, 1993.

    Article  MathSciNet  Google Scholar 

  9. M. X. Goemans and Y. Myung. A catalog of Steiner tree formulations. Networks, 23:19–28, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24(2):296–317, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. X. Goemans and D. P. Williamson. The primal-dual method for approximation algorithms and its application to network design problem. In D. S. Hochbaum, editor, Approximation Algorithms for NP-hard Problems. PWS, 1996.

    Google Scholar 

  12. F. K. Hwang, D. S. Richards, and P. Winter. The Steiner Tree Problem, volume 53 of Annals of Discrete Mathematics. North-Holland, Amsterdam, 1992.

    Google Scholar 

  13. P. N. Klein. A data structure for bicategories, with application to speeding up an approximation algorithm. Information Processing Letters, 52(6):303–307, 1994.

    Article  MATH  Google Scholar 

  14. T. Koch and A. Martin. SteinLib. ftp://ftp.zib.de/pub/Packages/mp-testdata/steinlib/index.html, 1997.

  15. T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32:207–232, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  16. Mehlhorn. A faster approximation algorithm for the Steiner problem in graphs. Information Processing Letters, 27:125–128, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  17. T. Polzin and S. Vahdati Daneshmand. A comparison of Steiner tree relaxations. Technical Report 5/1998, Universität Mannheim, 1998. (to appear in Discrete Applied Mathematics).

    Google Scholar 

  18. T. Polzin and S. Vahdati Daneshmand. Improved Algorithms for the Steiner Problem in Networks. Technical Report 06/1998, Universität Mannheim, 1998. (to appear in Discrete Applied Mathematics).

    Google Scholar 

  19. T. Polzin and S. Vahdati Daneshmand. Primal-Dual Approaches to the Steiner Problem. Technical Report 14/2000, Universität Mannheim, 2000.

    Google Scholar 

  20. S. Rajagopalan and V. V. Vazirani. On the bidirected cut relaxation for the metric Steiner tree problem. In Proceedings of the 10th ACM-SIAM Symposium on Discrete Algorithms, pages 742–751, 1999.

    Google Scholar 

  21. S. Vo\. Steiner’s problem in graphs: Heuristic methods. Discrete Applied Mathematics, 40:45–72, 1992.

    Article  MathSciNet  Google Scholar 

  22. R. T. Wong. A dual ascent approach for Steiner tree problems on a directed graph. Mathematical Programming, 28:271–287, 1984.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretische Informatik, Universität Mannheim, 68131, Mannheim, Germany

    Tobias Polzin & Siavash Vahdati

Authors
  1. Tobias Polzin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Siavash Vahdati
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Christian-Albrechts-Universität zu Kiel Institut für Informatik und praktische Mathematik Olshausenstr, 40, 24098, Kiel, Germany

    Klaus Jansen

  2. Computer Science Department, University of Maryland, A.V. Williams Bldg College Park, MD 20742, USA

    Samir Khuller

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Polzin, T., Vahdati, S. (2000). Primal-Dual Approaches to the Steiner Problem. In: Jansen, K., Khuller, S. (eds) Approximation Algorithms for Combinatorial Optimization. APPROX 2000. Lecture Notes in Computer Science, vol 1913. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44436-X_22

Download citation

  • DOI: https://doi.org/10.1007/3-540-44436-X_22

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67996-7

  • Online ISBN: 978-3-540-44436-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation