Document Open Access Logo

An Approximation Algorithm for Two-Edge-Connected Subgraph Problem via Triangle-Free Two-Edge-Cover

Authors Yusuke Kobayashi , Takashi Noguchi

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2023.49.pdf
  • Filesize: 0.62 MB
  • 10 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Combinatorial optimization
Keywords
  • approximation algorithm
  • survivable network design
  • minimum 2-edge-connected spanning subgraph
  • triangle-free 2-matching

Metrics

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

Abstract

The 2-Edge-Connected Spanning Subgraph problem (2-ECSS) is one of the most fundamental and well-studied problems in the context of network design. We are given an undirected graph G, and the objective is to find a 2-edge-connected spanning subgraph H of G with the minimum number of edges. For this problem, a lot of approximation algorithms have been proposed in the literature. In particular, very recently, Garg, Grandoni, and Ameli gave an approximation algorithm for 2-ECSS with a factor of 1.326, which is the best approximation ratio. In this paper, under the assumption that a maximum triangle-free 2-matching can be found in polynomial time in a graph, we give a (1.3+ε)-approximation algorithm for 2-ECSS, where ε is an arbitrarily small positive fixed constant. Note that a complicated polynomial-time algorithm for finding a maximum triangle-free 2-matching is announced by Hartvigsen in his PhD thesis, but it has not been peer-reviewed or checked in any other way. In our algorithm, we compute a minimum triangle-free 2-edge-cover in G with the aid of the algorithm for finding a maximum triangle-free 2-matching. Then, with the obtained triangle-free 2-edge-cover, we apply the arguments by Garg, Grandoni, and Ameli.

Cite As Get BibTex

Yusuke Kobayashi and Takashi Noguchi. An Approximation Algorithm for Two-Edge-Connected Subgraph Problem via Triangle-Free Two-Edge-Cover. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 49:1-49:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.49

Author Details

Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Takashi Noguchi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan

Funding

This work was partially supported by the joint project of Kyoto University and Toyota Motor Corporation, titled "Advanced Mathematical Science for Mobility Society", and by JSPS KAKENHI Grant Numbers JP20K11692 and JP22H05001.

References

  1. Joseph Cheriyan, András Sebő, and Zoltán Szigeti. Improving on the 1.5-approximation of a smallest 2-edge connected spanning subgraph. SIAM Journal on Discrete Mathematics, 14(2):170-180, 2001. URL: https://doi.org/10.1137/S0895480199362071.
  2. Joseph Cheriyan and Ramakrishna Thurimella. Approximating minimum-size k-connected spanning subgraphs via matching. SIAM Journal on Computing, 30(2):528-560, 2000. URL: https://doi.org/10.1137/S009753979833920X.
  3. Gérard Cornuéjols and William Pulleyblank. A matching problem with side conditions. Discrete Mathematics, 29(2):135-159, 1980. URL: https://doi.org/10.1016/0012-365x(80)90002-3.
  4. Artur Czumaj and Andrzej Lingas. On approximability of the minimum-cost k-connected spanning subgraph problem. In Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1999), pages 281-290, 1999. Google Scholar
  5. Cristina G Fernandes. A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem. Journal of Algorithms, 28(1):105-124, 1998. URL: https://doi.org/10.1006/jagm.1998.0931.
  6. Harold N. Gabow and Suzanne R. Gallagher. Iterated rounding algorithms for the smallest k-edge connected spanning subgraph. SIAM Journal on Computing, 41(1):61-103, 2012. URL: https://doi.org/10.1137/080732572.
  7. Harold N. Gabow, Michel X. Goemans, Éva Tardos, and David P. Williamson. Approximating the smallest k-edge connected spanning subgraph by LP-rounding. Networks, 53(4):345-357, 2009. URL: https://doi.org/10.1002/net.20289.
  8. Mohit Garg, Fabrizio Grandoni, and Afrouz Jabal Ameli. Improved approximation for two-edge-connectivity. In Proceedings of the 34th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), pages 2368-2410, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch92.
  9. Fabrizio Grandoni, Afrouz Jabal Ameli, and Vera Traub. Breaching the 2-approximation barrier for the forest augmentation problem. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2022), pages 1598-1611, 2022. URL: https://doi.org/10.1145/3519935.3520035.
  10. David Hartvigsen. Extensions of Matching Theory. PhD thesis, Carnegie Mellon University, 1984. Available at URL: https://david-hartvigsen.net.
  11. David Hartvigsen and Yanjun Li. Polyhedron of triangle-free simple 2-matchings in subcubic graphs. Mathematical Programming, 138:43-82, 2013. Google Scholar
  12. Christoph Hunkenschröder, Santosh Vempala, and Adrian Vetta. A 4/3-approximation algorithm for the minimum 2-edge connected subgraph problem. ACM Transactions on Algorithms, 15(4):1-28, 2019. URL: https://doi.org/10.1145/3341599.
  13. Kamal Jain. A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica, 21:39-60, 1998. URL: https://doi.org/10.1007/s004930170004.
  14. Samir Khuller and Uzi Vishkin. Biconnectivity approximations and graph carvings. Journal of the ACM, 41(2):214-235, 1994. URL: https://doi.org/10.1145/174652.174654.
  15. Yusuke Kobayashi. A simple algorithm for finding a maximum triangle-free 2-matching in subcubic graphs. Discrete Optimization, 7:197-202, 2010. URL: https://doi.org/10.1016/j.disopt.2010.04.001.
  16. Yusuke Kobayashi. Weighted triangle-free 2-matching problem with edge-disjoint forbidden triangles. Mathematical Programming, 192(1):675-702, 2022. URL: https://doi.org/10.1007/s10107-021-01661-y.
  17. Katarzyna Paluch and Mateusz Wasylkiewicz. A simple combinatorial algorithm for restricted 2-matchings in subcubic graphs - via half-edges. Information Processing Letters, 171:106146, 2021. URL: https://doi.org/10.1016/j.ipl.2021.106146.
  18. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency, volume 24 of Algorithms and Combinatorics. Springer-Verlag, Berlin, 2003. Google Scholar
  19. András Sebő and Jens 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, 34(5):597-629, 2014. URL: https://doi.org/10.1007/s00493-014-2960-3.
  20. Vera Traub and Rico Zenklusen. A better-than-2 approximation for weighted tree augmentation. In Proceedings of the IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS 2021), pages 1-12, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00010.
  21. Vera Traub and Rico Zenklusen. A (1.5+ε)-approximation algorithm for weighted connectivity augmentation. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC 2023), STOC 2023, pages 1820-1833, 2023. URL: https://doi.org/10.1145/3564246.3585122.
  22. Santosh Vempala and Adrian Vetta. Factor 4/3 approximations for minimum 2-connected subgraphs. In Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX 2000), pages 262-273, 2000. URL: https://doi.org/10.1007/3-540-44436-X_26.
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