Skip to main content
Log in

On better heuristics for Steiner minimum trees

  • Published:
Mathematical Programming Submit manuscript

Abstract

Finding a shortest network interconnecting a given set of points in a metric space is called the Steiner minimum tree problem. The Steiner ratio is the largest lower bound for the ratio between lengths of a Steiner minimum tree and a minimum spanning tree for the same set of points. In this paper, we show that in a metric space, if the Steiner ratio is less than one and finding a Steiner minimum tree for a set of size bounded by a fixed number can be performed in polynomial time, then there exists a polynomialtime heuristic for the Steiner minimum tree problem with performance ratio bigger than the Steiner ratio. It follows that in the Euclidean plane, there exists a polynomial-time heuristic for Steiner minimum trees with performance ratio bigger than\({\textstyle{1 \over 2}}\sqrt 3 \). This solves a long-standing open problem.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Berman and V. Ramaiye, “An approximation algorithm for the Steiner tree problem,” manuscript.

  2. M. Bern, “Two probabilistic results on rectilinear Steiner trees,”Proceedings of the 18th Annual ACM Symposium on Theory of Computing (1986) 433–441.

  3. F.R.K. Chung and R.L. Graham, “A new bound for euclidean Steiner minimum trees,”Annual New York Academy of Sciences 440 (1985) 328–346.

    Google Scholar 

  4. F.R.K. Chung and F.K. Hwang, “A lower bound for the Steiner tree problem,”SIAM Journal of Applied Mathematics 34 (1978) 27–36.

    Google Scholar 

  5. D.-Z. Du and F.K. Hwang, “A new bound for the Steiner ratio,”Transactions of American Mathematical Society 278 (1983) 137–148.

    Google Scholar 

  6. D.-Z. Du and F.K. Hwang, “An approach for proving lower bounds: solution of Gilbert—Pollak's conjecture on Steiner ratio,”Proceedings of the 31st Annual Symposium on Foundations of Computer Science (1990) 76–85.

  7. L.R. Foulds and R.L. Graham, “The Steiner problem in phylogeny is NP-complete,”Advances in Applied Mathematics 3 (1982) 43–49.

    Google Scholar 

  8. M.R. Garey, R.L. Graham and D.S. Johnson, “The complexity of computing Steiner minimal trees,”SIAM Journal of Applied Mathematics 32 (1977) 835–859.

    Google Scholar 

  9. M.R. Garey and D.S. Johnson, “The rectilinear Steiner tree problem is NP-complete,”SIAM Journal of Applied Mathematics 32 (1977) 826–834.

    Google Scholar 

  10. E.N. Gilbert and H.O. Pollak, “Steiner minimal trees,”SIAM Journal of Applied Mathematics 16 (1968) 1–29.

    Google Scholar 

  11. R.L. Graham and F.K. Hwang, “Remarks on Steiner minimal trees,”Bulletin of Institute of Mathematics, Academia Sinica 4 (1976) 177–182.

    Google Scholar 

  12. F.K. Hwang, “On Steiner minimal trees with rectilinear distance,”SIAM Journal of Applied Mathematics 30 (1976) 104–114.

    Google Scholar 

  13. F.K. Hwang and Y.C. Yao, “Comments on Bern's probabilistic results on rectilinear Steiner trees,”Algorithmica 5 (1990) 591–598.

    Google Scholar 

  14. R.M. Karp, “Reducibility among combinatorial problems,” in: R.E. Miller and J.W. Tatcher, ed.,Complexity of Computer Computation (Plenum Press, New York, 1972) pp. 85–103.

    Google Scholar 

  15. H.O. Pollak, “Some remarks on the Steiner problem,”Journal of Combinatorial Theory Series A 24 (1978) 278–295.

    Google Scholar 

  16. J.H. Rubinstein and D.A. Thomas, “The Steiner ratio conjecture for six points,” to appear in:Journal of Combinatoria Theory Series A.

  17. A.Z. Zelikovsky, “The 11/6-approximation algorithm for the Steiner problem on networks,” manuscript.

Download references

Author information

Authors and Affiliations

Authors

Additional information

Part of this work was done while this author visited the Department of Computer Science, Princeton University, supported in part by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, under NSF grant STC88-09648, supported in part by NSF grant No. CCR-8920505, and also supported in part by the National Natural Science Foundation of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Du, DZ., Zhang, Y. On better heuristics for Steiner minimum trees. Mathematical Programming 57, 193–202 (1992). https://doi.org/10.1007/BF01581080

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01581080

Key words

Navigation