Skip to main content

Netzwerk-Design-Probleme

  • Chapter
  • First Online:
Kombinatorische Optimierung

Part of the book series: Masterclass ((MASTERCLASS))

  • 6241 Accesses

Zusammenfassung

Wie kann man günstige Teilgraphen mit bestimmten Zusammenhangseigenschaften konstruieren? Das Steinerbaumproblem ist der bekannteste Fall; wir geben den (In 4 + ϵ) – Approximationsalgorithmus an. Für allgemeines Netzwerkdesign betrachten wir einen primal-dualen Approximationsalgorithmus und Jains Algorithmus mit iterativem Runden. Ein anderes Problem, das als VPN-Problem bekannt ist, kann sogar optimal gelöst werden.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 44.99
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.

Literatur

Allgemeine Literatur:

  • Cheng, X., und Du, D.-Z. [2001]: Steiner Trees in Industry. Kluwer, Dordrecht 2001

    Google Scholar 

  • Du, D.-Z., Smith, J.M., und Rubinstein, J.H. [2000]: Advances in Steiner Trees. Kluwer, Boston 2000

    Google Scholar 

  • Goemans, M.X., und 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, Hrsg.), PWS, Boston, 1996

    Google Scholar 

  • Grötschel, M., Monma, C.L., und 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, Hrsg.), Elsevier, Amsterdam 1995

    Google Scholar 

  • Gupta, A., und Könemann, J. [2011]: Approximation algorithms for network design: a survey. Surveys in Operations Research and Management Science 16 (2011) 3–20

    Article  Google Scholar 

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

    Google Scholar 

  • Kerivin, H., und Mahjoub, A.R. [2005]: Design of survivable networks: a survey. Networks 46 (2005), 1–21

    Article  MathSciNet  Google Scholar 

  • Lau, L.C., Ravi, R., und Singh, M. [2011]: Iterative Methods in Combinatorial Optimization. Cambridge University Press 2011, Kapitel 10

    Google Scholar 

  • Prömel, H.J., und Steger, A. [2002]: The Steiner Tree Problem. Vieweg, Braunschweig 2002

    Book  Google Scholar 

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

    Chapter  Google Scholar 

  • Vazirani, V.V. [2001]: Approximation Algorithms. Springer, Berlin 2001, Kapitel 22 und 23

    Google Scholar 

Zitierte Literatur:

  • Agrawal, A., Klein, P.N., und 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

    Google Scholar 

  • Arora, S. [1998]: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM 45 (1998), 753–782

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Bertsimas, D., und 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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Björklund, A., Husfeldt, T., Kaski, P., und Koivisto, M. [2007]: Fourier meets Möbius: fast subset convolution. Proceedings of the 39th Annual ACM Symposium on Theory of Computing (2007), 67–74

    Google Scholar 

  • Borchers, A., und Du, D.-Z. [1997]: The k-Steiner ratio in graphs. SIAM Journal on Computing 26 (1997), 857–869

    Article  MathSciNet  Google Scholar 

  • Borradaile, G., Klein, P., und Mathieu, C., [2009]: An O(n log n) approximation scheme for Steiner tree in planar graphs. ACM Transactions on Algorithms 5 (2009), Article 31

    Google Scholar 

  • Byrka, J., Grandoni, F., Rothvoß, T., und Sanità, L. [2013]: Steiner tree approximation via iterative randomized rounding. Journal of the ACM 60 (2013), Artikel 6

    Article  MathSciNet  Google Scholar 

  • Chakraborty, T., Chuzhoy, J., und Khanna, S. [2008]: Network design for vertex connectivity. Proceedings of the 40th Annual ACM Symposium on Theory of Computing (2008), 167–176

    Google Scholar 

  • Cheriyan, J., und Vetta, A. [2007]: Approximation algorithms for network design with metric costs. SIAM Journal on Discrete Mathematics 21 (2007), 612–636

    Article  MathSciNet  Google Scholar 

  • Chlebík, M., und Chlebíková, J. [2008]: The Steiner tree problem on graphs: Inapproximability results. Theoretical Computer Science 406 (2008), 207–214

    Google Scholar 

  • Choukhmane, E. [1978]: Une heuristique pour le problème de l’arbre de Steiner. RAIRO Recherche Opérationnelle 12 (1978), 207–212 [auf Französisch]

    Google Scholar 

  • Chuzhoy, J., und Khanna, S. [2009]: An O(k3 log n)-approximation algorithm for vertexconnectivity survivable network design. Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (2009), 437–441

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Du, D.-Z., Zhang, Y., und Feng, Q. [1991]: On better heuristic for Euclidean Steiner minimum trees. Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (1991), 431–439 (siehe auch Mathematical Programming 57 (1992) 193–202)

    Google Scholar 

  • Erickson, R.E., Monma, C.L., und 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  Google Scholar 

  • Fleischer, L., Jain, K., und Williamson, D.P. [2006]: Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems. Journal of Computer and System Sciences 72 (2006), 838–867

    Article  MathSciNet  Google Scholar 

  • Fuchs, B., Kern, W., Mölle, D., Richter, S., Rossmanith, P., und Wang, X. [2007]: Dynamic programming for minimum Steiner trees. Theory of Computing Systems 41 (2007), 493–500

    Article  MathSciNet  Google Scholar 

  • Gabow, H.N. [2005]: An improved analysis for approximating the smallest k-edge connected spanning subgraph of a multigraph. SIAM Journal on Discrete Mathematics 19 (2005), 1–18

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Gabow, H.N., Goemans, M.X., Tardos, É., und Williamson, D.P. [2009]: Approximating the smallest k-edge connected spanning subgraph by LP-rounding. Networks 53 (2009), 345–357

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Gilbert, E.N., und Pollak, H.O. [1968]: Steiner minimal trees. SIAM Journal on Applied Mathematics 16 (1968), 1–29

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Goemans, M.X., Goldberg, A.V., Plotkin, S., Shmoys, D.B., Tardos, É., undWilliamson, 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., Olver, N., Rothvoß, T., und Zenklusen, R. [2012]: Matroids and integrality gaps for hypergraphic Steiner tree relaxations. Proceedings of the 44th Annual ACM Symposium on Theory of Computing (2012), 1161–1176

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Goyal, N., Olver, N., und Shepherd, F.B. [2013]: The VPN conjecture is true. Journal of the ACM 60 (2013), Artikel 17

    Article  MathSciNet  Google Scholar 

  • Grandoni, F., Kaibel, V., Oriolo, G., und Skutella, M. [2008]: A short proof of the VPN tree routing conjecture on ring networks. Operations Research Letters 36 (2008), 361–365

    Article  MathSciNet  Google Scholar 

  • Gröpl, C., Hougardy, S., Nierhoff, T., und Prömel, H.J. [2001]: Approximation algorithms for the Steiner tree problem in graphs. In: Cheng und Du [2001], S. 235–279

    Google Scholar 

  • Gupta, A., Kleinberg, J., Kumar, A., Rastogi, R., und Yener, B. [2001]: Provisioning a virtual private network: a network design problem for multicommodity flow. Proceedings of the 33th Annual ACM Symposium on Theory of Computing (2001), 389–398

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Hetzel, A. [1995]: Verdrahtung im VLSI-Design: Spezielle Teilprobleme und ein sequentielles Lösungsverfahren. Dissertation, Universität Bonn, 1995

    Google Scholar 

  • Hougardy, S., und 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 

  • Hougardy, S., Silvanus, J., und Vygen, J. [2017]: Dijkstra meets Steiner: a fast exact goaloriented Steiner tree algorithm. Mathematical Programming Computation 9 (2017), 135–202

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Jain, K. [2001]: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21 (2001), 39–60

    Article  MathSciNet  Google Scholar 

  • Jothi, R., Raghavachari, B., und Varadarajan, S. [2003]: A 5/4-approximation algorithm for minimum 2-edge-connectivity. Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (2003), 725–734

    Google Scholar 

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

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  • Könemann, J., Olver, N., Pashkovish, K., Ravi, R., Swamy, C., und Vygen, J. [2017]: On the integrality gap of the prize-collecting Steiner forest LP. Proceedings of APPROX 2017, Artikel 17

    Google Scholar 

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

    Google Scholar 

  • Kortsarz, G., Krauthgamer, R., und Lee, J.R. [2004]: Hardness of approximation for vertexconnectivity network design problems. SIAM Journal on Computing 33 (2004), 704–720

    Article  MathSciNet  Google Scholar 

  • Kou, L.T. [1990]: On efficient implementation of an approximation algorithm for the Steiner tree problem. Acta Informatica 27 (1990), 369–380

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Melkonian, V., und Tardos, É. [2004]: Algorithms for a network design problem with crossing supermodular demands. Networks 43 (2004), 256–265

    Google Scholar 

  • Nagarajan, V., Ravi, R., und Singh, M. [2010]: Simpler analysis of LP extreme points for traveling salesman and survivable network design problems. Operations Research Letters 38 (2010), 156–160

    Article  MathSciNet  Google Scholar 

  • Robins, G., und Zelikovsky, A. [2005]: Tighter bounds for graph Steiner tree approximation. SIAM Journal on Discrete Mathematics 19 (2005), 122–134

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  • Vygen, J. [2011]: Faster algorithm for optimum Steiner trees. Information Processing Letters, 111 (2011), 1075–1079

    Article  MathSciNet  Google Scholar 

  • Warme, D.M., Winter, P., und Zachariasen, M. [2000]: Exact algorithms for plane Steiner tree problems: a computational study. In: Advances in Steiner trees (D.-Z. Du, J.M. Smith, J.H. Rubinstein, Hrsg.), Kluwer Academic Publishers, Boston, 2000, S. 81–116

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bernhard Korte .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Korte, B., Vygen, J. (2018). Netzwerk-Design-Probleme. In: Kombinatorische Optimierung. Masterclass. Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57691-5_20

Download citation

Publish with us

Policies and ethics