Minimum Weight 2-Edge-Connected Spanning Subgraphs in Planar Graphs

  • André Berger
  • Michelangelo Grigni
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

Abstract

We present a linear time algorithm exactly solving the 2-edge connected spanning subgraph (2-ECSS) problem in a graph of bounded treewidth. Using this with Klein’s diameter reduction technique [15], we find a linear time PTAS for the problem in unweighted planar graphs, and the first PTAS for the problem in weighted planar graphs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arora, S., Grigni, M., Karger, D., Klein, P., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: Proceedings of the Ninth Annual ACM-SIAM SODA, pp. 33–41. ACM Press, New York (1998)Google Scholar
  2. 2.
    Arora, S.: Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In: Proceedings of the Thirtyseventh Annual FOCS, pp. 2–11. IEEE Computer Society Press, Los Alamitos (1996)Google Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Berger, A., Grigni, M., Zhao, H.: A well-connected separator for planar graphs. Technical Report TR-2004-026-A, Emory University (2004)Google Scholar
  5. 5.
    Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Borradaile, G., Kenyon-Mathieu, C., Klein, P.: A Polynomial-Time Approximation Scheme for Steiner Tree in Planar Graphs. In: Proceedings of the Eighteenth Annual ACM-SIAM SODA, ACM Press, New York (2007)Google Scholar
  7. 7.
    Csaba, B., Karpinski, M., Krysta, P.: Approximability of dense and sparse instances of minimum 2-connectivity, TSP and path problems. In: Proceedings of the 13th Annual ACM-SIAM Symposium On Discrete Mathematics (SODA), pp. 74–83. ACM Press, New York (2002)Google Scholar
  8. 8.
    Czumaj, A., Grigni, M., Sissokho, P., Zhao, H.: Approximation schemes for minimum 2-edge-connected and biconnected subgraphs in planar graphs. In: Proceedings of the Fifteenth Annual ACM-SIAM SODA, pp. 489–498. ACM Press, New York (2004)Google Scholar
  9. 9.
    Czumaj, A., Lingas, A.: On approximability of the minimumcost k-connected spanning subgraph problem. In: Proceedings of the Tenth Annual ACM-SIAM SODA, pp. 281–290. ACM Press, New York (1999)Google Scholar
  10. 10.
    Demaine, E.D., Hajiaghayi, M.: Fast algorithms for hard graph problems: Bidimensionality, minors, and local treewidth. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 517–533. Springer, Heidelberg (2005)Google Scholar
  11. 11.
    Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. Journal of Graph Algorithms and Applications 3(3), 1–27 (1999)MathSciNetGoogle Scholar
  12. 12.
    Jothi, R., Raghavachari, B., Varadarajan, S.: A 5/4-approximation algorithm for minimum 2-edge-connectivity. In: Proceedings of the Fourteenth Annual ACM-SIAM SODA, pp. 725–734. ACM Press, New York (2003)Google Scholar
  13. 13.
    Khuller, S.: Approximation algorithms for finding highly connected subgraphs. In: Dorit, S. (ed.) Approximation Algorithms for NP-hard Problems, PWS Publishing Company (1996)Google Scholar
  14. 14.
    Khuller, S., Vishkin, U.: Biconnectivity approximations and graph carvings. Journal of the ACM 41(2), 214–235 (1994)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Klein, P.N.: A linear-time approximation scheme for planar weighted TSP. In: Proceedings of the 46th Annual IEEE FOCS, pp. 647–657. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  16. 16.
    Klein, P.N.: A subset spanner for planar graphs, with application to subset TSP. In: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing (STOC), pp. 749–756. ACM Press, New York (2006)CrossRefGoogle Scholar
  17. 17.
    Kloks, T. (ed.): Treewidth. LNCS, vol. 842. Springer, Berlin (1994)MATHGoogle Scholar
  18. 18.
    Rao, S.B., Smith, W.D.: Approximating geometrical graphs via “spanners” and “banyans”. In: STOC 1998. Proceedings of the Thirtieth annual ACM Symposium on Theory of Computing, pp. 540–550. ACM Press, New York (1998)CrossRefGoogle Scholar
  19. 19.
    Röhrig, H.: Tree decomposition: A feasibility study. Master’s thesis, Max-Planck-Institut für Informatik in Saarbrücken (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • André Berger
    • 1
  • Michelangelo Grigni
    • 2
  1. 1.Department of Mathematics, Technical University Berlin, Axel-Springer-Str. 54A, 10623 BerlinGermany
  2. 2.Department of Mathematics and Computer Science, Emory University, Atlanta GA 30322USA

Personalised recommendations