Skip to main content
SpringerLink
Log in

The Two-Edge Connectivity Survivable Network Problem in Planar Graphs

  • Conference paper
Automata, Languages and Programming (ICALP 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5125))

Included in the following conference series:

Abstract

Consider the following problem: given a graph with edge-weights and a subset Q of vertices, find a minimum-weight subgraph in which there are two edge-disjoint paths connecting every pair of vertices in Q. The problem is a failure-resilient analog of the Steiner tree problem, and arises in telecommunications applications. A more general formulation, also employed in telecommunications optimization, assigns a number (or requirement) r v  ∈ {0,1,2} to each vertex v in the graph; for each pair u,v of vertices, the solution network is required to contain min{r u , r v } edge-disjoint u-to-v paths.

We address the problem in planar graphs, considering a popular relaxation in which the solution is allowed to use multiple copies of the input-graph edges (paying separately for each copy). The problem is SNP-hard in general graphs and NP-hard in planar graphs. We give the first polynomial-time approximation scheme in planar graphs. The running time is O(n logn).

Under the additional restriction that the requirements are in {0,2} for vertices on the boundary of a single face of a planar graph, we give a linear-time algorithm to find the optimal solution.

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 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.00
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.

References

  1. Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berger, A., Czumaj, A., Grigni, M., Zhao, H.: Approximation schemes for minimum 2-connected spanning subgraphs in weighted planar graphs. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 472–483. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  3. Berger, A., Grigni, M.: Minimum weight 2-edge-connected spanning subgraphs in planar graphs. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 90–101. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  4. Borradaile, G., Kenyon-Mathieu, C., Klein, P.: A polynomial-time approximation scheme for Steiner tree in planar graphs. In: 18th SODA, pp. 1285–1294 (2007)

    Google Scholar 

  5. Borradaile, G., Klein, P., Mathieu, C.: Steiner tree in planar graphs: An O(n logn) approximation scheme with singly exponential dependence on epsilon. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 275–286. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  6. Czumaj, A., Lingas, A.: On approximability of the minimum cost k-connected spanning subgraph problem. In: 10th SODA, pp. 281–290 (1999)

    Google Scholar 

  7. Erickson, R., Monma, C., Veinott, A.: Send-and-split method for minimum-concave-cost network flows. Math. Op. Res. 12, 634–664 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eswaran, K., Tarjan, R.: Augmentation problems. SIAM J. Comput. 5(4), 653–665 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  9. Frederickson, G., Jájá, J.: Approximation algorithms for several graph augmentation problems. SIAM J. Comput. 10(2), 270–283 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  10. Goemans, M., Goldberg, A., Plotkin, S., Shmoys, D., Tardos, É., Williamson, D.: Improved approximation algorithms for network design problems. In: 5th SODA, pp. 223–232 (1994)

    Google Scholar 

  11. Henzinger, M., Klein, P., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. J. Comput. System Sci. 55(1), 3–23 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jothi, R., Raghavachari, B., Varadarajan, S.: A 5/4-approximation algorithm for minimum 2-edge-connectivity. In: 14th SODA, pp. 725–734 (2003)

    Google Scholar 

  14. Khuller, S., Vishkin, U.: Biconnectivity approximations and graph carvings. J. ACM 41(2), 214–235 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Klein, P.: A linear-time approximation scheme for planar weighted TSP. In: 46th FOCS, pp. 647–647 (2005)

    Google Scholar 

  16. Klein, P.: A subset spanner for planar graphs, with application to subset TSP. In: 38th STOC, pp. 749–756 (2006)

    Google Scholar 

  17. Klein, P., Ravi, R.: When cycles collapse: A general approximation technique for constraind two-connectivity problems. In: 3rd IPCO, pp. 39–55 (1993)

    Google Scholar 

  18. Ravi, R.: Approximation algorithms for Steiner augmentations for two-connectivity. Technical Report TR-CS-92-21, Brown University (1992)

    Google Scholar 

  19. Resende, M., Pardalos, P. (eds.): Handbook of Optimization in Telecommunications. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  20. Williamson, D., Goemans, M., Mihail, M., Vazirani, V.: A primal-dual approximation algorithm for generalized Steiner network problems. In: 35th STOC, pp. 708–717 (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Combinatorics and Optimization, University of Waterloo, Canada

    Glencora Borradaile

  2. Computer Science Department, Brown University, USA

    Philip Klein

Authors
  1. Glencora Borradaile
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philip Klein
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavík, Iceland

    Luca Aceto

  2. Department of Computer Science, IT-Parken, University of Aarhus, Åbogade 34, 8200, ÅrhusN, Denmark

    Ivan Damgård

  3. Department of Computer Science, University of Liverpool, Ashton Building, L69 3BX, Liverpool, UK

    Leslie Ann Goldberg

  4. School of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavik, Iceland

    Magnús M. Halldórsson

  5. Department of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavík, Iceland

    Anna Ingólfsdóttir

  6. Université de Bordeaux-1, LaBRI, 351, Cours de la Libération, 33405, Talence cedex, France

    Igor Walukiewicz

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Borradaile, G., Klein, P. (2008). The Two-Edge Connectivity Survivable Network Problem in Planar Graphs. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_40

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-70575-8_40

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70574-1

  • Online ISBN: 978-3-540-70575-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation