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Approximation Algorithms for Steiner Connectivity Augmentation

Authors Daniel Hathcock , Michael Zlatin

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Author Details

Daniel Hathcock
  • Carnegie Mellon University, Pittsburgh, PA, USA
Michael Zlatin
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Hathcock, Daniel: Supported by the NSF Graduate Research Fellowship grant DGE-2140739.

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Daniel Hathcock and Michael Zlatin. Approximation Algorithms for Steiner Connectivity Augmentation. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.67

Abstract

We consider connectivity augmentation problems in the Steiner setting, where the goal is to augment the edge-connectivity between a specified subset of terminal nodes. In the Steiner Augmentation of a Graph problem (k-SAG), we are given a k-edge-connected subgraph H of a graph G. The goal is to augment H by including links from G of minimum cost so that the edge-connectivity between nodes of H increases by 1. This is a generalization of the Weighted Connectivity Augmentation Problem, in which only links between pairs of nodes in H are available for the augmentation. In the Steiner Connectivity Augmentation Problem (k-SCAP), we are given a Steiner k-edge-connected graph connecting terminals R, and we seek to add links of minimum cost to create a Steiner (k+1)-edge-connected graph for R. Note that k-SAG is a special case of k-SCAP. The results of Ravi, Zhang and Zlatin for the Steiner Tree Augmentation problem yield a (1.5+ε)-approximation for 1-SCAP and for k-SAG when k is odd [Ravi et al., 2023]. In this work, we give a (1 + ln{2} +ε)-approximation for the Steiner Ring Augmentation Problem (SRAP). This yields a polynomial time algorithm with approximation ratio (1 + ln{2} + ε) for 2-SCAP. We obtain an improved approximation guarantee for SRAP when the ring consists of only terminals, yielding a (1.5+ε)-approximation for k-SAG for any k.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Approximation Algorithms
  • Steiner Connectivity
  • Network Design

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References

  1. Al Borchers and Ding-Zhu Du. Thek-steiner ratio in graphs. SIAM Journal on Computing, 26(3):857-869, 1997. URL: https://doi.org/10.1137/S0097539795281086.
  2. Federica Cecchetto, Vera Traub, and Rico Zenklusen. Bridging the gap between tree and connectivity augmentation: unified and stronger approaches. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 370-383, 2021. URL: https://doi.org/10.1145/3406325.3451086.
  3. Joseph Cheriyan and Zhihan Gao. Approximating (unweighted) tree augmentation via lift-and-project, part I: stemless TAP. Algorithmica, 80(2):530-559, 2018. URL: https://doi.org/10.1007/s00453-016-0270-4.
  4. Joseph Cheriyan and Zhihan Gao. Approximating (unweighted) tree augmentation via lift-and-project, part II. Algorithmica, 80(2):608-651, 2018. URL: https://doi.org/10.1007/s00453-017-0275-7.
  5. Joseph Cheriyan, Howard Karloff, Rohit Khandekar, and Jochen Könemann. On the integrality ratio for tree augmentation. Operations Research Letters, 36(4):399-401, 2008. URL: https://doi.org/10.1016/j.orl.2008.01.009.
  6. Nachshon Cohen and Zeev Nutov. A (1 + ln 2)-approximation algorithm for minimum-cost 2-edge-connectivity augmentation of trees with constant radius. Theoret. Comput. Sci., 489/490:67-74, 2013. URL: https://doi.org/10.1016/j.tcs.2013.04.004.
  7. Yefim Dinitz and Alek Vainshtein. The connectivity carcass of a vertex subset in a graph and its incremental maintenance. In Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pages 716-725, 1994. URL: https://doi.org/10.1145/195058.195442.
  8. Stuart E Dreyfus and Robert A Wagner. The steiner problem in graphs. Networks, 1(3):195-207, 1971. URL: https://doi.org/10.1002/net.3230010302.
  9. Tamás Fleiner and András Frank. A quick proof for the cactus representation of mincuts. EGRES Quick Proof, 3:2009, 2009. Google Scholar
  10. Greg N Frederickson and Joseph Ja’Ja’. Approximation algorithms for several graph augmentation problems. SIAM Journal on Computing, 10(2):270-283, 1981. URL: https://doi.org/10.1137/0210019.
  11. Waldo Gálvez, Fabrizio Grandoni, Afrouz Jabal Ameli, and Krzysztof Sornat. On the cycle augmentation problem: hardness and approximation algorithms. Theory of Computing Systems, 65(6):985-1008, 2021. URL: https://doi.org/10.1007/s00224-020-10025-6.
  12. Mohit Garg, Fabrizio Grandoni, and Afrouz Jabal Ameli. Improved approximation for two-edge-connectivity. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 2368-2410. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch92.
  13. Fabrizio Grandoni, Afrouz Jabal Ameli, and Vera Traub. Breaching the 2-approximation barrier for the forest augmentation problem. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1598-1611. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520035.
  14. Christoph Hunkenschröder, Santosh S. Vempala, and Adrian Vetta. A 4/3-approximation algorithm for the minimum 2-edge connected subgraph problem. ACM Trans. Algorithms, 15(4):55:1-55:28, 2019. URL: https://doi.org/10.1145/3341599.
  15. Kamal Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21(1):39-60, 2001. URL: https://doi.org/10.1007/s004930170004.
  16. Yusuke Kobayashi and Takashi Noguchi. An approximation algorithm for two-edge-connected subgraph problem via triangle-free two-edge-cover. In Satoru Iwata and Naonori Kakimura, editors, 34th International Symposium on Algorithms and Computation, ISAAC 2023, December 3-6, 2023, Kyoto, Japan, volume 283 of LIPIcs, pages 49:1-49:10. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2023.49.
  17. Guy Kortsarz, Robert Krauthgamer, and James R Lee. Hardness of approximation for vertex-connectivity network design problems. SIAM Journal on Computing, 33(3):704-720, 2004. URL: https://doi.org/10.1137/S0097539702416736.
  18. Zeev Nutov. Approximating steiner networks with node-weights. SIAM Journal on Computing, 39(7):3001-3022, 2010. URL: https://doi.org/10.1137/080729645.
  19. Ojas Parekh, R. Ravi, and Michael Zlatin. On small-depth tree augmentations. CoRR, abs/2111.00148, 2021. URL: https://doi.org/10.48550/arXiv.2111.00148.
  20. R Ravi, Weizhong Zhang, and Michael Zlatin. Approximation algorithms for steiner tree augmentation problems. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2429-2448. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch94.
  21. András Sebö 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. Comb., 34(5):597-629, 2014. URL: https://doi.org/10.1007/s00493-014-2960-3.
  22. Vera Traub and Rico Zenklusen. A (1.5+$$1epsilon)-approximation algorithm for weighted connectivity augmentation. arXiv preprint arXiv:2209.07860, 2022. Google Scholar
  23. Vera Traub and Rico Zenklusen. A better-than-2 approximation for weighted tree augmentation. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 1-12, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00010.
  24. Vera Traub and Rico Zenklusen. Local search for weighted tree augmentation and steiner tree. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3253-3272. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.128.
  25. David P Williamson, Michel X Goemans, Milena Mihail, and Vijay V Vazirani. A primal-dual approximation algorithm for generalized steiner network problems. In Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 708-717, 1993. URL: https://doi.org/10.1145/167088.167268.
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