Mathematical Programming

, Volume 142, Issue 1–2, pp 133–167 | Cite as

The Steiner connectivity problem

  • Ralf Borndörfer
  • Marika Karbstein
  • Marc E. Pfetsch
Full Length Paper Series A


The Steiner connectivity problem has the same significance for line planning in public transport as the Steiner tree problem for telecommunication network design. It consists in finding a minimum cost set of elementary paths to connect a subset of nodes in an undirected graph and is, therefore, a generalization of the Steiner tree problem. We propose an extended directed cut formulation for the problem which is, in comparison to the canonical undirected cut formulation, provably strong, implying, e.g., a class of facet defining Steiner partition inequalities. Since a direct application of this formulation is computationally intractable for large instances, we develop a partial projection method to produce a strong relaxation in the space of canonical variables that approximates the extended formulation. We also investigate the separation of Steiner partition inequalities and give computational evidence that these inequalities essentially close the gap between undirected and extended directed cut formulation. Using these techniques, large Steiner connectivity problems with up to 900 nodes can be solved within reasonable optimality gaps of typically less than five percent.

Mathematics Subject Classification

90C10 90C27 90C57 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Achterberg T.: SCIP: solving constraint integer programs. Math. Program Comput. 1(1), 1–41 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Balakrishnan A., Mangnanti T.L., Mirchandani P.: Network design. In: Dell’Amico, M., Maffioli, F., Martello, S. (eds) Annotated Bibliographies in Combinatorial Optimization, chap. 18, pp. 311–334. Wiley, Chichester (1997)Google Scholar
  3. 3.
    Balas E., Ceria S., Cornuéjols G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Prog. 58, 295–324 (1993)CrossRefMATHGoogle Scholar
  4. 4.
    Balas E., Ng S.M.: On the set covering polytope: I. All the facets with coefficients in {0,1,2}. Math. Program. 43, 57–69 (1989)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Borndörfer R., Grötschel M., Pfetsch M.E.: A column-generation approach to line planning in public transport. Transp. Sci. 41(1), 123–132 (2007)CrossRefGoogle Scholar
  6. 6.
    Borndörfer, R., Neumann, M.: Linienoptimierung—reif für die Praxis? ZIB-Report 10-20. Zuse Institute Berlin (2010)Google Scholar
  7. 7.
    Borndörfer, R., Neumann, M.: Models for Line Planning with Transfers. ZIB-Report 10-11, Zuse Institute Berlin (2010)Google Scholar
  8. 8.
    Borndörfer, R., Neumann, M., Pfetsch, M.E.: The Steiner Connectivity Problem. ZIB-Report 09-07, Zuse Institute Berlin (2009).
  9. 9.
    Bussieck, M.: Gams—lop.gms: line optimization.
  10. 10.
    Bussieck, M.R.: Optimal Lines in Public Rail Transport. PhD thesis, TU Braunschweig (1997)Google Scholar
  11. 11.
    Bussieck M.R., Kreuzer P., Zimmermann U.T.: Optimal lines for railway systems. Eur. J. Oper. Res. 96(1), 54–63 (1997)CrossRefMATHGoogle Scholar
  12. 12.
    Bussieck, M.R., Lindner, T., Lübbecke, M.E.: A fast algorithm for near optimal line plans. Math. Methods Oper. Res. 59(2) (2004)Google Scholar
  13. 13.
    Chopra S., Rao M.: The Steiner tree problem I: formulations, compositions and extension of facets. Math. Programm. 64(2), 209–229 (1994)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Claessens M.T., van Dijk N.M., Zwaneveld P.J.: Cost optimal allocation of rail passanger lines. Eur. J. Oper. Res. 110(3), 474–489 (1998)CrossRefMATHGoogle Scholar
  15. 15.
    Conforti M., Cornuéjols G., Zambelli G.: Extended formulations in combinatorial optimization. 4OR 8, 1–48 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cornuéjols, G.: Combinatorial Optimization: Packing and Covering. CBMS-NSF regional conference series in applied mathematics, vol. 74. SIAM, Berlin (2001)Google Scholar
  17. 17.
    Dix, A.: Das statische Linienplanungsproblem. Diploma thesis, TU Berlin (2007)Google Scholar
  18. 18.
    Duin C.W.: Steiner’s Problem in Graphs. PhD thesis, University of Amsterdam (1993)Google Scholar
  19. 19.
    Feige, U.: A threshold of ln n for approximating set-cover. In: Proceedings of the 28th ACM Symposium on Theory of Computing, pp. 314–318 (1996)Google Scholar
  20. 20.
    Feldman, J., Ruhl, M.: The directed steiner network problem is tractable for a constant number of terminals. In: IEEE Symposium on Foundations of Computer Science, pp. 299–308 (1999)Google Scholar
  21. 21.
    Giandomenico M., Letchford A.N., Rossi F., Smriglio S.: An application of the Lovász-Schrijver M(K, K) operator to the stable set problem. Math. Programm. 120(2), 381–401 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Grötschel M., Lovász L., Schrijver A.: Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, Berlin (1988)CrossRefGoogle Scholar
  23. 23.
    Grötschel M., Monma C.L.: Integer polyhedra arising from certain network design problems with connectivity constraints. SIAM J. Discret. Math. 3(4), 502–523 (1990)CrossRefMATHGoogle Scholar
  24. 24.
    Grötschel M., Monma C.L., Stoer M.: Computational results with a cutting plane algorithm for designing communication networks with low-connectivity constraints. Oper. Res. 40, 309–330 (1992)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Günlük O.: A branch-and-cut algorithm for capacitated network design problems. Math. Programm. 86, 17–39 (1999)CrossRefMATHGoogle Scholar
  26. 26.
  27. 27.
    Koch T., Martin A.: Solving Steiner tree problems in graphs to optimality. Networks 32, 207–232 (1998)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Lovász L., Schrijver A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1, 166–190 (1991)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Nachtigall, K., Jerosch, K.: Simultaneous network line planning and traffic assignment. In: Fischetti, M., Widmayer, P. (eds.) ATMOS 2008—8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems, Dagstuhl, Germany. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Germany (2008)Google Scholar
  30. 30.
    Pochet Y., Wolsey L.A.: Production Planning by Mixed Integer Programming. Springer, New York (2006)MATHGoogle Scholar
  31. 31.
    Polzin, T.: Algorithms for the Steiner Problems in Networks. PhD thesis, University of Saarland, Saarbrücken (2003)Google Scholar
  32. 32.
    Polzin T., Daneshmand S.V.: A comparison of Steiner tree relaxations. Discret. Appl. Math. 112, 241–261 (2001)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Prömel Hans J., Steger A.: The Steiner Tree Problem. Vieweg, Braunschweig (2002)CrossRefMATHGoogle Scholar
  34. 34.
    Raghavan S., Magnanti T.L.: Network connectivity. In: Mauro, D., Maffioli, F., Martello, S. (eds) Annotated Bibliographies in Combinatorial Optimization, pp. 335–354. Wiley, Chichester (1997)Google Scholar
  35. 35.
    Schöbel, A., Scholl, S.: Line planning with minimal traveling time. In: Kroon Leo, G., Möhring Rolf, H. (eds.) 5th Workshop on Algorithmic Methods and Models for Optimization of Railways, Dagstuhl, Germany. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2006)Google Scholar
  36. 36.
    Scholl, S.: Customer-Oriented Line Planning. PhD thesis, Universität Göttingen (2005)Google Scholar
  37. 37.
    SCIP—Solving Constraint Integer Programs.
  38. 38.
    Transportation network test problems.
  39. 39.
    Torres, L.M., Torres, R., Borndörfer, R., Pfetsch, M.E.: Line planning on paths and tree networks with applications to the Quito Trolebús System. Int. Trans. Oper. Res. 18(455–472) (2011)Google Scholar
  40. 40.
    Vanderbeck F., Wolsey L.: Reformulation and decomposition of integer programs. In: Jünger, M., Liebling, T., Naddef, D., Nemhauser, G.L., Pulleyblank, W., Reinelt, G., Rinaldi, G., Wolsey, L. (eds) 50 Years of Integer Programming 1958–2008, chap. 13, pp. 431–502. Springer, Berlin (2010)Google Scholar

Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  • Ralf Borndörfer
    • 1
  • Marika Karbstein
    • 1
  • Marc E. Pfetsch
    • 2
  1. 1.Zuse Institute BerlinBerlinGermany
  2. 2.Department of Mathematics, Discrete OptimizationTU DarmstadtDarmstadtGermany

Personalised recommendations