Skip to main content
SpringerLink
Log in

Network Design Problems

  • Chapter
Combinatorial Optimization

Part of the book series: Algorithms and Combinatorics ((AC,volume 21))

Abstract

Connectivity is a very important concept in combinatorial optimization. In Chapter 8 we showed how to compute the connectivity between each pair of vertices of an undirected graph. Now we are looking for subgraphs that satisfy certain connectivity requirements.

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 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

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

General Literature:

  • Hwang, F.K., Richards, D.S., and Winter, P. [ 1992 ]: The Steiner Tree Problem; Annals of Discrete Mathematics 53. North-Holland, Amsterdam 1992

    Google Scholar 

  • Goemans, M.X., and Williamson, D.P. [ 1996 ]: The primal-dual method for approximation algorithms and its application to network design problems. In: Approximation Algorithms for NP-Hard Problems. ( D.S. Hochbaum, ed.), PWS, Boston, 1996

    Google Scholar 

  • Grötschel, M., Monma, C.L., and Stoer, M. [ 1995 ]: Design of survivable networks. In: Handbooks in Operations Research and Management Science; Volume 7; Network Models ( M.O. Ball, T.L. Magnanti, C.L. Monma, G.L. Nemhauser, eds.), Elsevier, Amsterdam 1995

    Google Scholar 

  • Stoer, M. [ 1992 ]: Design of Survivable Networks. Springer, Berlin 1992

    MATH  Google Scholar 

  • Vazirani, V.V. [ 2000 ]: Approximation Algorithms. Springer, forthcoming.

    Google Scholar 

Cited References:

  • Agrawal, A., Klein, P., and Ravi, R. [ 1995 ]: When trees collide: an approximation algorithm for the generalized Steiner tree problem in networks. SIAM Journal on Computing 24 (1995), 440–456

    Article  MathSciNet  MATH  Google Scholar 

  • Arora, S. [ 1996 ]: Polynomial time approximation schemes for Euclidean TSP and other geometric problems. Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science (1996), 2–12

    Google Scholar 

  • Arora, S. [ 1997 ]: Nearly linear time approximation schemes for Euclidean TSP and other geometric problems. Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science (1997), 554–563

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Bern, M., and Plassmann, P. [ 1989 ]: The Steiner problem with edge lengths 1 and 2. Information Processing Letters 32 (1989), 171–176

    Article  MathSciNet  MATH  Google Scholar 

  • Bertsimas, D., and Teo, C. [ 1995 ]: From valid inequalities to heuristics: a unified view of primal-dual approximation algorithms in covering problems. Operations Research 46 (1998), 503–514

    MathSciNet  Google Scholar 

  • Bertsimas, D., and Teo, C. [ 1997 ]: The parsimonious property of cut covering problems and its applications. Operations Research Letters 21 (1997), 123–132

    Article  MathSciNet  MATH  Google Scholar 

  • Carr, R., and Ravi, R. [1998]: A new bound for the 2-edge connected subgraph problem. Proceedings of the 6th Conference on Integer Programming and Combinatorial Optimization; LNCS 1412 (R.E. Bixby, E.A. Boyd, R.Z. Rios-Mercado, eds.), Springer, Berlin 1998, 112–125

    Google Scholar 

  • Cheriyan, J., and Thurimella, R. [ 1996 ]: Approximating minimum-size k-connected spanning subgraphs via matching. Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science (1996), 292–301

    Google Scholar 

  • Cheriyan, J., Sebö, A., and Szigeti, Z. [1998]: An improved approximation algorithm for minimum size 2-edge connected spanning subgraphs. Proceedings of the 6th Conference on Integer Programming and Combinatorial Optimization; LNCS 1412 (R.E. Bixby, E.A. Boyd, R.Z. Rios-Mercado, eds.), Springer, Berlin 1998, 126–136

    Google Scholar 

  • Clementi, A.E.F., and Trevisan, L. [ 1996 ]: Improved non-approximability results for minimum vertex cover with density constraints. In: Computing and Combinatorics, Second International Conference COCOON ‘86; LNCS 1090 ( J.-Y. Cai, C.K. Wong, eds.), Springer, Berlin 1996

    Google Scholar 

  • Dreyfus, S.E., and Wagner, R.A. [ 1972 ]: The Steiner problem in graphs. Networks 1 (1972), 195–207

    MathSciNet  MATH  Google Scholar 

  • Du, D.-Z., and Hwang, F.K. [ 1992 ]: A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica 7 (1992), 121–135

    Article  MathSciNet  MATH  Google Scholar 

  • Erickson, R.E., Monma, C.L., and Veinott, A.F., Jr. [ 1987 ]: Send-and-split method for minimum concave-cost network flows. Mathematics of Operations Research 12 (1987), 634–664

    Article  MathSciNet  MATH  Google Scholar 

  • Fernandez, C.G. [ 1998 ]: A better approximation ratio for the minimum size k-edge- connected spanning subgraph problem. Journal of Algorithms 28 (1998), 105–124

    Article  MathSciNet  Google Scholar 

  • Gabow, H.N., Goemans, M.X., Williamson, D.P. [ 1998 ]: An efficient approximation algorithm for the survivable network design problem. Mathematical Programming B 82 (1998), 13–40

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Garey, M.R., and Johnson, D.S. [ 1977 ]: The rectilinear Steiner tree problem is NP-complete. SIAM Journal on Applied Mathematics 32 (1977), 826–834

    Article  MathSciNet  MATH  Google Scholar 

  • Goemans, M.X., and Bertsimas, D.J. [ 1993 ]: Survivable networks, linear programming and the parsimonious property, Mathematical Programming 60 (1993), 145–166

    Article  MathSciNet  MATH  Google Scholar 

  • Goemans, M.X., Goldberg, A.V., Plotkin, S., Shmoys, D.B., Tardos, E., and Williamson, D.P. [ 1994 ]: Improved approximation algorithms for network design problems. Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms (1994), 223–232

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Hanan, M. [ 1966 ]: On Steiner’s problem with rectlinear distance. SIAM Journal on Applied Mathematics 14 (1966), 255–265

    Article  MathSciNet  MATH  Google Scholar 

  • Hetzel, A. [ 1995 ]: Verdrahtung im VLSI-Design: Spezielle Teilprobleme and ein sequentielles Lösungsverfahren. Ph.D. thesis, University of Bonn, 1995

    Google Scholar 

  • Hougardy, S., and Prömel, H.J. [ 1999 ]: A 1.598 approximation algorithm for the Steiner tree problem in graphs. Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (1999), 448–453

    Google Scholar 

  • Hwang, F.K. [ 1976 ]: On Steiner minimal trees with rectilinear distance. SIAM Journal on Applied Mathematics 30 (1976), 104–114

    Article  MathSciNet  MATH  Google Scholar 

  • Jain, K. [ 1998 ]: A factor 2 approximation algorithm for the generalized Steiner network problem. Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (1998), 448–457, submitted to Combinatorica

    Google Scholar 

  • Karp, R.M. [1972]: Reducibility among combinatorial problems. In: Complexity of Computer Computations ( R.E. Miller, J.W. Thatcher, eds.), Plenum Press, New York 1972, pp. 85–103

    Chapter  Google Scholar 

  • Karpinski, M., and Zelikovsky, A. [ 1997 ]: New approximation algorithms for Steiner tree problems. Journal of Combinatorial Optimization 1 (1997), 47–65

    Article  MathSciNet  MATH  Google Scholar 

  • Khuller, S., and Raghavachari, B. [ 1996 ]: Improved approximation algorithms for uniform connectivity problems. Journal of Algorithms 21 (1996), 434–450

    Article  MathSciNet  MATH  Google Scholar 

  • Khuller, S., and Vishkin, U. [1994]: Biconnectivity augmentations and graph carvings. Journal of the ACM 41 (1994), 214–235

    Article  MathSciNet  MATH  Google Scholar 

  • Klein, P., and Ravi, R. [ 1993 ]: When cycles collapse: a general approximation technique for constrained two-connectivity problems. Proceedings of the 3rd MPS Conference on Integer Programming and Combinatorial Optimization (1993), 39–55

    Google Scholar 

  • Korte, B., Prömel, H.J., and Steger, A. [1990]: Steiner trees in VLSI-layout. In: Paths, Flows, and VLSI-Layout ( B. Korte, L. Lovâsz, H.J. Prömel, A. Schrijver, eds.), Springer, Berlin 1990, pp. 185–214

    Google Scholar 

  • Kou, L. [ 1990 ]: A faster approximation algorithm for the Steiner problem in graphs. Acta Informatica 27 (1990), 369–380

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Martin, A. [ 1992 ]: Packen von Steinerbäumen: Polyedrische Studien und Anwendung. Ph.D. thesis, Technical University of Berlin 1992 [in German]

    Google Scholar 

  • Mehlhorn, K. [ 1988 ]: A faster approximation algorithm for the Steiner problem in graphs. Information Processing Letters 27 (1988), 125–128

    Article  MathSciNet  MATH  Google Scholar 

  • Melkonian, V., and Tardos, É. [1999]: Approximation algorithms for a directed network design problem. Proceedings of the 7th Conference on Integer Programming and Combinatorial Optimization; LNCS 1610 (G. Cornuéjols, R.E. Burkard, G.J. Woeginger, eds.), Springer, Berlin 1999, pp. 345–360

    Google Scholar 

  • Ravi, R., and Williamson, D.P. [ 1997 ]: An approximation algorithm for minimum-cost vertex-connectivity problems. Algorithmica 18 (1997), 21–43

    Article  MathSciNet  MATH  Google Scholar 

  • Robins, G., and Zelikovsky, A. [ 2000 ]: Improved Steiner tree approximation in graphs. Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (2000), 770–779

    Google Scholar 

  • Takahashi, M., and Matsuyama, A. [ 1980 ]: An approximate solution for the Steiner problem in graphs. Mathematica Japonica 24 (1980), 573–577

    MathSciNet  MATH  Google Scholar 

  • Warme, D.M., Winter, P., and Zachariasen, M. [ 1999 ]: Exact algorithms for plane Steiner tree problems: a computational study. In: Advances in Steiner trees ( D.-Z. Du, J.M. Smith, J.H. Rubinstein, eds.), Kluwer Academic Publishers, Boston, to appear

    Google Scholar 

  • Williamson, D.P., Goemans, M.X., Mihail, M., and Vazirani, V.V. [ 1995 ]: A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica 15 (1995), 435–454

    Article  MathSciNet  MATH  Google Scholar 

  • Zelikovsky, A.Z. [ 1993 ]: An 11/6-approximation algorithm for the network Steiner problem. Algorithmica 9 (1993), 463–470

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute for Discrete Mathematics, University of Bonn, Lennéstraße 2, 53113, Bonn, Germany

    Bernhard Korte & Jens Vygen

Authors
  1. Bernhard Korte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jens Vygen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Korte, B., Vygen, J. (2000). Network Design Problems. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21708-5_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-21708-5_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-21710-8

  • Online ISBN: 978-3-662-21708-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation