Document Open Access Logo

A PTAS for Three-Edge-Connected Survivable Network Design in Planar Graphs

Authors Glencora Borradaile, Baigong Zheng

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2017.3.pdf
  • Filesize: 0.57 MB
  • 13 pages

Document Identifiers

Subject Classification

Keywords
  • Three-Edge Connectivity
  • Polynomial-Time Approximation Schemes
  • Planar Graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We consider the problem of finding the minimum-weight subgraph that satisfies given connectivity requirements. Specifically, given a requirement r in {0, 1, 2, 3} for every vertex, we seek the minimum-weight subgraph that contains, for every pair of vertices u and v, at least min{r(v), r(u)} edge-disjoint u-to-v paths. We give a polynomial-time approximation scheme (PTAS) for this problem when the input graph is planar and the subgraph may use multiple copies of any given edge (paying for each edge separately). This generalizes an earlier result for r in {0, 1, 2}. In order to achieve this PTAS, we prove some properties of triconnected planar graphs that may be of independent interest.

Cite As Get BibTex

Glencora Borradaile and Baigong Zheng. A PTAS for Three-Edge-Connected Survivable Network Design in Planar Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 3:1-3:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.3

Author Details

Glencora Borradaile
Baigong Zheng

References

  1. S. Arora, M. Grigni, D. Karger, P. Klein, and A. Woloszyn. A polynomial-time approximation scheme for weighted planar graph TSP. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 33-41, 1998. Google Scholar
  2. B. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM, 41(1):153-180, 1994. URL: http://dx.doi.org/10.1145/174644.174650.
  3. M. Bateni, M. Hajiaghayi, and D. Marx. Approximation schemes for Steiner forest on planar graphs and graphs of bounded treewidth. J. ACM, 58(5):21, 2011. URL: http://dx.doi.org/10.1145/2027216.2027219.
  4. A. Berger, A. Czumaj, M. Grigni, and H. Zhao. Approximation schemes for minimum 2-connected spanning subgraphs in weighted planar graphs. In Proceedings of the 13th European Symposium on Algorithms, volume 3669 of Lecture Notes in Computer Science, pages 472-483, 2005. Google Scholar
  5. A. Berger and M. Grigni. Minimum weight 2-edge-connected spanning subgraphs in planar graphs. In Proceedings of the 34th International Colloquium on Automata, Languages and Programming, volume 4596 of Lecture Notes in Computer Science, pages 90-101, 2007. URL: http://dx.doi.org/10.1007/978-3-540-73420-8_10.
  6. G. Borradaile, E. Demaine, and S. Tazari. Polynomial-time approximation schemes for subset-connectivity problems in bounded-genus graphs. Algorithmica, 2012. Online. URL: http://dx.doi.org/10.1016/j.jda.2012.04.011.
  7. G. Borradaile, C. Kenyon-Mathieu, and P. Klein. A polynomial-time approximation scheme for Steiner tree in planar graphs. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, volume 7, pages 1285-1294, 2007. Google Scholar
  8. G. Borradaile and P. Klein. The two-edge connectivity survivable network problem in planar graphs. In Proceedings of the 35th International Colloquium on Automata, Languages and Programming, pages 485-501, 2008. Google Scholar
  9. G. Borradaile and P. Klein. The two-edge connectivity survivable-network design problem in planar graphs. ACM Transactions on Algorithms, 12(3):30:1-30:29, 2016. Google Scholar
  10. G. Borradaile, P. Klein, and C. Mathieu. An O(n log n) approximation scheme for Steiner tree in planar graphs. ACM Transactions on Algorithms, 5(3):1-31, 2009. Google Scholar
  11. G. Borradaile and B. Zheng. A PTAS for three-edge-connected survivable network design in planar graphs. CoRR, abs/1611.03889, 2016. URL: http://arxiv.org/abs/1611.03889.
  12. A. Czumaj and A. Lingas. A polynomial time approximation scheme for euclidean minimum cost k-connectivity. In Automata, Languages and Programming, pages 682-694. Springer, 1998. Google Scholar
  13. A. Czumaj and A. Lingas. On approximability of the minimum cost k-connected spanning subgraph problem. In Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 281-290, 1999. Google Scholar
  14. R. Erickson, C. Monma, and A. Veinott. Send-and-split method for minimum-concave-cost network flows. Mathematics of Operations Research, 12:634-664, 1987. Google Scholar
  15. K. Eswaran and R. Tarjan. Augmentation problems. SIAM Journal on Computing, 5(4):653-665, 1976. Google Scholar
  16. G. Frederickson and J. Jájá. Approximation algorithms for several graph augmentation problems. SIAM Journal on Computing, 10(2):270-283, 1981. Google Scholar
  17. D. A. Holton, B. Jackson, A. Saito, and N. C. Wormald. Removable edges in 3-connected graphs. J. Graph Theory, 14:465-475, 1990. Google Scholar
  18. K. Jain. A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica, 2001(1):39-60, 21. Google Scholar
  19. S. Khuller and U. Vishkin. Biconnectivity approximations and graph carvings. Journal of the ACM, 41(2):214-235, 1994. Google Scholar
  20. P. Klein. A subset spanner for planar graphs, with application to subset TSP. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 749-756, 2006. URL: http://dx.doi.org/10.1145/1132516.1132620.
  21. P. Klein. A linear-time approximation scheme for TSP in undirected planar graphs with edge-weights. SIAM Journal on Computing, 37(6):1926-1952, 2008. Google Scholar
  22. P. Klein, C. Mathieu, and H. Zhou. Correlation clustering and two-edge-connected augmentation for planar graphs. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), volume 30 of Leibniz International Proceedings in Informatics (LIPIcs), pages 554-567. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015. Google Scholar
  23. P. Klein and R. Ravi. When cycles collapse: A general approximation technique for constraind two-connectivity problems. In Proceedings of the 3rd International Conference on Integer Programming and Combinatorial Optimization, pages 39-55, 1993. Google Scholar
  24. A. Sebő and J. Vygen. Shorter tours by nicer ears: 7/5-approximation for the graph-tsp, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. Combinatorica, pages 1-34, 2014. Google Scholar
  25. H. Whitney. Non-separable and planar graphs. Trans. Amer. Math. Soc., 34:339-362, 1932. Google Scholar
  26. B. Zheng. Linear-time approximation schemes for planar minimum three-edge connected and three-vertex connected spanning subgraphs. CoRR, abs/1701.08315, 2017. URL: http://arxiv.org/abs/1701.08315.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact