How Many Steiner Terminals Can You Connect in 20 Years?

  • Ralf BorndörferEmail author
  • Nam-Dũng Hoang
  • Marika Karbstein
  • Thorsten Koch
  • Alexander Martin


Steiner trees are constructed to connect a set of terminal nodes in a graph. This basic version of the Steiner tree problem is idealized, but it can effectively guide the search for successful approaches to many relevant variants, from both a theoretical and a computational point of view. This article illustrates the theoretical and algorithmic progress on Steiner tree type problems on two examples, the Steiner connectivity and the Steiner tree packing problem.


Greedy Algorithm Steiner Tree Steiner Tree Problem Grid Graph Node Disjoint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank an anonymous referee and the editors for helpful comments and suggestions that improved the presentation of this paper. The work of Marika Karbstein was supported by the DFG Research Center Matheon “Mathematics for key technologies”.


  1. 1.
    Achterberg, T., Raack, C.: The MCF-separator—detecting and exploiting multi-commodity flows in MIPs. Math. Program. Comput. 2, 125–165 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Boit, C.: Personal communication (2004) Google Scholar
  3. 3.
    Borndörfer, R., Karbstein, M.: A direct connection approach to integrated line planning and passenger routing. In: Delling, D., Liberti, L. (eds.) 12th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. OpenAccess Series in Informatics (OASIcs), vol. 25, pp. 47–57. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Wadern (2012) Google Scholar
  4. 4.
    Borndörfer, R., Karbstein, M., Pfetsch, M.E.: The Steiner connectivity problem. Math. Program., Ser. A (2012). doi: 10.1007/s10107-012-0564-5 Google Scholar
  5. 5.
    Brady, M.L., Brown, D.J.: VLSI routing: four layers suffice. In: Preparata, F.P. (ed.) Advances in Computing Research: VLSI Theory, vol. 2, pp. 245–258. Jai Press, London (1984) Google Scholar
  6. 6.
    Burstein, M., Pelavin, R.: Hierarchical wire routing. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 2, 223–234 (1983) CrossRefGoogle Scholar
  7. 7.
    Chopra, S.: Comparison of formulations and a heuristic for packing Steiner trees in a graph. Ann. Oper. Res. 50, 143–171 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Coohoon, J.P., Heck, P.L.: BEAVER: a computational-geometry-based tool for switchbox routing. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 7, 684–697 (1988) CrossRefGoogle Scholar
  10. 10.
    Feige, U.: A threshold of lnn for approximating set-cover. In: Proceedings of the 28th ACM Symposium on Theory of Computing, pp. 314–318 (1996) Google Scholar
  11. 11.
    Frank, A.: Connections in Combinatorial Optimization. Oxford University Press, Oxford (2011) zbMATHGoogle Scholar
  12. 12.
    Fulkerson, D.R.: Blocking and anti-blocking pairs of polyhedra. Math. Program. 1, 168–194 (1971) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Grötschel, M., Jünger, M., Reinelt, G.: Via minimization with pin preassignments and layer preference. Z. Angew. Math. Mech. 69(11), 393–399 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Grötschel, M., Martin, A., Weismantel, R.: Optimum path packing on wheels: the consecutive case. Comput. Math. Appl. 31, 23–35 (1996) zbMATHCrossRefGoogle Scholar
  15. 15.
    Grötschel, M., Martin, A., Weismantel, R.: Packing Steiner trees: a cutting plane algorithm and computational results. Math. Program. 72, 125–145 (1996) zbMATHCrossRefGoogle Scholar
  16. 16.
    Grötschel, M., Martin, A., Weismantel, R.: Packing Steiner trees: further facets. Eur. J. Comb. 17, 39–52 (1996) zbMATHCrossRefGoogle Scholar
  17. 17.
    Grötschel, M., Martin, A., Weismantel, R.: Packing Steiner trees: polyhedral investigations. Math. Program. 72, 101–123 (1996) zbMATHCrossRefGoogle Scholar
  18. 18.
    Grötschel, M., Martin, A., Weismantel, R.: Packing Steiner trees: separation algorithms. SIAM J. Discrete Math. 9, 233–257 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Grötschel, M., Martin, A., Weismantel, R.: The Steiner tree packing problem in VLSI design. Math. Program. 78(2), 265–281 (1997) zbMATHCrossRefGoogle Scholar
  20. 20.
    Held, S., Korte, B., Rautenbach, D., Vygen, J.: Combinatorial optimization in VLSI design. In: Chvátal, V. (ed.) Combinatorial Optimization—Methods and Applications. NATO Science for Peace and Security Series—D: Information and Communication Security, vol. 31, pp. 33–96 (2011) Google Scholar
  21. 21.
    Hoàng, N.D., Koch, T.: Steiner tree packing revisited. Math. Methods Oper. Res. 76(1), 95–123 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Jørgensen, D.G., Meyling, M.: Application of column generation techniques in VLSI design. Master’s thesis, Department of Computer Science, University of Copenhagen (2000) Google Scholar
  23. 23.
    Karbstein, M.: Line planning and connectivity. Ph.D. thesis, TU Berlin (2013) Google Scholar
  24. 24.
    Koch, T.:. ZIMPL.
  25. 25.
    Koch, T.: Rapid mathematical programming. Ph.D. thesis, Technische Universität Berlin (2004) Google Scholar
  26. 26.
    Korte, B., Prömel, H.J., Steger, A.: Steiner trees in VLSI-layout. In: Korte, B., Lovász, L., Prömel, H.J., Schrijver, A. (eds.) Paths, Flows, and VLSI-Layout. Springer, Berlin (1990) Google Scholar
  27. 27.
    Lengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. Wiley, New York (1990) zbMATHGoogle Scholar
  28. 28.
    Lipski, W.: On the structure of three-layer wireable layouts. In: Preparata, F.P. (ed.) Advances in Computing Research: VLSI Theory, vol. 2, pp. 231–244. Jai Press, London (1984) Google Scholar
  29. 29.
    Luk, W.K.: A greedy switch-box router. Integration 3, 129–149 (1985) Google Scholar
  30. 30.
    Martin, A.: Packen von Steinerbäumen: Polyedrische Studien und Anwendungen. Ph.D. thesis, Technische Universität Berlin (1992) Google Scholar
  31. 31.
    Oxley, J.G.: Matroid Theory. Oxford University Press, Oxford (1992) zbMATHGoogle Scholar
  32. 32.
    Polzin, T.: Algorithms for the Steiner problem in networks. Ph.D. thesis, Universität des Saarlandes (2003) Google Scholar
  33. 33.
    Prömel, H., Steger, A.: The Steiner Tree Problem. Vieweg, Wiesbaden (2002) zbMATHCrossRefGoogle Scholar
  34. 34.
    Raack, C., Koster, A.M.C.A., Orlowski, S., Wessäly, R.: On cut-based inequalities for capacitated network design polyhedra. Networks 57(2), 141–156 (2011) MathSciNetzbMATHGoogle Scholar
  35. 35.
    Raghavan, S., Magnanti, T.: Network connectivity. In: Dell’Amico, M., Maffioli, F., Martello, S. (eds.) Annotated Bibliographies in Combinatorial Optimization, pp. 335–354. Wiley, Chichester (1997) Google Scholar
  36. 36.
    Robacker, J.T.: Min-Max theorems on shortest chains and disjunct cuts of a network. Research Memorandum RM-1660, The RAND Corporation, Santa Monica, CA (1956) Google Scholar
  37. 37.
    Wolsey, L.A.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2(4), 385–393 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Wong, R.T.: A dual ascent approach for Steiner tree problems on a directed graph. Math. Program. 28, 271–287 (1984) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ralf Borndörfer
    • 1
    Email author
  • Nam-Dũng Hoang
    • 2
  • Marika Karbstein
    • 1
  • Thorsten Koch
    • 1
  • Alexander Martin
    • 3
  1. 1.Konrad-Zuse-Zentrum für Informationstechnik BerlinBerlinGermany
  2. 2.Faculty of Mathematics, Mechanics, and InformaticsVietnam National UniversityHanoiVietnam
  3. 3.Department MathematikFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

Personalised recommendations