Document Open Access Logo

A Constant-Factor Approximation for Quasi-Bipartite Directed Steiner Tree on Minor-Free Graphs

Authors Zachary Friggstad, Ramin Mousavi

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2023.13.pdf
  • Filesize: 0.76 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Routing and network design problems
Keywords
  • Directed Steiner tree
  • Combinatorial optimization
  • approximation algorithms

Metrics

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

Abstract

We give the first constant-factor approximation algorithm for quasi-bipartite instances of Directed Steiner Tree on graphs that exclude fixed minors. In particular, for K_r-minor-free graphs our approximation guarantee is O(r⋅√(log r)) and, further, for planar graphs our approximation guarantee is 20.
Our algorithm uses the primal-dual scheme. We employ a more involved method of determining when to buy an edge while raising dual variables since, as we show, the natural primal-dual scheme fails to raise enough dual value to pay for the purchased solution. As a consequence, we also demonstrate integrality gap upper bounds on the standard cut-based linear programming relaxation for the Directed Steiner Tree instances we consider.

Cite As Get BibTex

Zachary Friggstad and Ramin Mousavi. A Constant-Factor Approximation for Quasi-Bipartite Directed Steiner Tree on Minor-Free Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.13

Author Details

Zachary Friggstad
  • Department of Computing Science, University of Alberta, Canada
Ramin Mousavi
  • Department of Computing Science, University of Alberta, Canada

References

  1. MohammadHossein Bateni, MohammadTaghi Hajiaghayi, and Dániel Marx. Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth. Journal of the ACM (JACM), 58(5):1-37, 2011. Google Scholar
  2. Marshall Bern and Paul Plassmann. The steiner problem with edge lengths 1 and 2. Information Processing Letters, 32(4):171-176, 1989. Google Scholar
  3. Glencora Borradaile, Philip Klein, and Claire Mathieu. An o (n log n) approximation scheme for steiner tree in planar graphs. ACM Transactions on Algorithms (TALG), 5(3):1-31, 2009. Google Scholar
  4. Jarosław Byrka, Fabrizio Grandoni, Thomas Rothvoß, and Laura Sanità. Steiner tree approximation via iterative randomized rounding. Journal of the ACM (JACM), 60(1):1-33, 2013. Google Scholar
  5. Gruia Calinescu and Alexander Zelikovsky. The polymatroid steiner problems. J. Combonatorial Optimization, 33(3):281-294, 2005. Google Scholar
  6. Deeparnab Chakrabarty, Nikhil R Devanur, and Vijay V Vazirani. New geometry-inspired relaxations and algorithms for the metric steiner tree problem. Mathematical programming, 130(1):1-32, 2011. Google Scholar
  7. Chun-Hsiang Chan, Bundit Laekhanukit, Hao-Ting Wei, and Yuhao Zhang. Polylogarithmic approximation algorithm for k-connected directed steiner tree on quasi-bipartite graphs. arXiv preprint, 2019. URL: https://arxiv.org/abs/1911.09150.
  8. Moses Charikar, Chandra Chekuri, To-Yat Cheung, Zuo Dai, Ashish Goel, Sudipto Guha, and Ming Li. Approximation algorithms for directed steiner problems. Journal of Algorithms, 33(1):73-91, 1999. Google Scholar
  9. Erik D Demaine, MohammadTaghi Hajiaghayi, and Philip N Klein. Node-weighted steiner tree and group steiner tree in planar graphs. ACM Transactions on Algorithms (TALG), 10(3):1-20, 2014. Google Scholar
  10. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 624-633, 2014. Google Scholar
  11. Jack Edmonds. Optimum branchings. Journal of Research of the national Bureau of Standards B, 71(4):233-240, 1967. Google Scholar
  12. Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634-652, 1998. Google Scholar
  13. Andreas Emil Feldmann, Jochen Könemann, Neil Olver, and Laura Sanità. On the equivalence of the bidirected and hypergraphic relaxations for steiner tree. Mathematical programming, 160(1):379-406, 2016. Google Scholar
  14. Zachary Friggstad, Jochen Könemann, Young Kun-Ko, Anand Louis, Mohammad Shadravan, and Madhur Tulsiani. Linear programming hierarchies suffice for directed steiner tree. In International Conference on Integer Programming and Combinatorial Optimization, pages 285-296. Springer, 2014. Google Scholar
  15. Zachary Friggstad, Jochen Könemann, and Mohammad Shadravan. A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs . In Rasmus Pagh, editor, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016), volume 53 of Leibniz International Proceedings in Informatics (LIPIcs), pages 3:1-3:11, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  16. Zachary Friggstad and Ramin Mousavi. A constant-factor approximation for quasi-bipartite directed steiner tree on minor-free graphs. arXiv preprint, 2021. URL: https://arxiv.org/abs/2111.02572.
  17. Zachary Friggstad and Ramin Mousavi. An O(log k)-Approximation for Directed Steiner Tree in Planar Graphs. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261 of Leibniz International Proceedings in Informatics (LIPIcs), pages 63:1-63:14, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Google Scholar
  18. Rohan Ghuge and Viswanath Nagarajan. Quasi-polynomial algorithms for submodular tree orienteering and other directed network design problems. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1039-1048. SIAM, 2020. Google Scholar
  19. Michel X Goemans, Neil Olver, Thomas Rothvoß, and Rico Zenklusen. Matroids and integrality gaps for hypergraphic steiner tree relaxations. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 1161-1176, 2012. Google Scholar
  20. Michel X Goemans and David P Williamson. The primal-dual method for approximation algorithms and its application to network design problems. Approximation algorithms for NP-hard problems, pages 144-191, 1997. Google Scholar
  21. Fabrizio Grandoni, Bundit Laekhanukit, and Shi Li. O (log2 k/log log k)-approximation algorithm for directed steiner tree: a tight quasi-polynomial-time algorithm. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 253-264, 2019. Google Scholar
  22. Sudipto Guha, Anna Moss, Joseph Naor, and Baruch Schieber. Efficient recovery from power outage. In Proceedings of the thirty-first annual ACM symposium on Theory of computing, pages 574-582, 1999. Google Scholar
  23. Eran Halperin and Robert Krauthgamer. Polylogarithmic inapproximability. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 585-594, 2003. Google Scholar
  24. Tomoya Hibi and Toshihiro Fujito. Multi-rooted greedy approximation of directed steiner trees with applications. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 215-224. Springer, 2012. Google Scholar
  25. Marek Karpinski and Alexander Zelikovsky. New approximation algorithms for the steiner tree problems. Journal of Combinatorial Optimization, 1(1):47-65, 1997. Google Scholar
  26. Jochen Könemann, Sina Sadeghian, and Laura Sanita. An lmp o (log n)-approximation algorithm for node weighted prize collecting steiner tree. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 568-577. IEEE, 2013. Google Scholar
  27. Shi Li and Bundit Laekhanukit. Polynomial integrality gap of flow lp for directed steiner tree. arXiv preprint, 2021. URL: https://arxiv.org/abs/2110.13350.
  28. Carsten Moldenhauer. Primal-dual approximation algorithms for node-weighted steiner forest on planar graphs. Information and Computation, 222:293-306, 2013. Google Scholar
  29. Christos H Papadimitriou and Kenneth Steiglitz. Combinatorial optimization: algorithms and complexity. Courier Corporation, 1998. Google Scholar
  30. Hans Jürgen Prömel and Angelika Steger. A new approximation algorithm for the steiner tree problem with performance ratio 5/3. Journal of Algorithms, 36(1):89-101, 2000. Google Scholar
  31. Sridhar Rajagopalan and Vijay V Vazirani. On the bidirected cut relaxation for the metric steiner tree problem. In SODA, volume 99, pages 742-751, 1999. Google Scholar
  32. Gabriel Robins and Alexander Zelikovsky. Tighter bounds for graph steiner tree approximation. SIAM Journal on Discrete Mathematics, 19(1):122-134, 2005. Google Scholar
  33. Thomas Rothvoß. Directed steiner tree and the lasserre hierarchy. arXiv preprint, 2011. URL: https://arxiv.org/abs/1111.5473.
  34. Andrew Thomason. The extremal function for complete minors. Journal of Combinatorial Theory, Series B, 81(2):318-338, 2001. Google Scholar
  35. Mikkel Thorup. Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM (JACM), 51(6):993-1024, 2004. Google Scholar
  36. Alexander Zelikovsky. A series of approximation algorithms for the acyclic directed steiner tree problem. Algorithmica, 18(1):99-110, 1997. Google Scholar
  37. Alexander Z Zelikovsky. An 11/6-approximation algorithm for the network steiner problem. Algorithmica, 9(5):463-470, 1993. Google Scholar
  38. Leonid Zosin and Samir Khuller. On directed steiner trees. In SODA, volume 2, pages 59-63. Citeseer, 2002. Google Scholar
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