An O(log k)-Approximation for Directed Steiner Tree in Planar Graphs

Authors Zachary Friggstad, Ramin Mousavi

Thumbnail PDF


  • Filesize: 1.19 MB
  • 14 pages

Document Identifiers

Author Details

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

Cite AsGet BibTex

Zachary Friggstad and Ramin Mousavi. An O(log k)-Approximation for Directed Steiner Tree in Planar Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We present an O(log k)-approximation for both the edge-weighted and node-weighted versions of Directed Steiner Tree in planar graphs where k is the number of terminals. We extend our approach to Multi-Rooted Directed Steiner Tree, in which we get a O(R+log k)-approximation for planar graphs for where R is the number of roots.

Subject Classification

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


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


  1. Ittai Abraham and Cyril Gavoille. Object location using path separators. In Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing, pages 188-197, 2006. Google Scholar
  2. 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
  3. Marshall Bern and Paul Plassmann. The steiner problem with edge lengths 1 and 2. Information Processing Letters, 32(4):171-176, 1989. Google Scholar
  4. 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
  5. 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
  6. Gruia Calinescu and Alexander Zelikovsky. The polymatroid steiner problems. J. Combonatorial Optimization, 33(3):281-294, 2005. Google Scholar
  7. 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
  8. Vincent Cohen-Addad. Bypassing the surface embedding: approximation schemes for network design in minor-free graphs. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 343-356, 2022. 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. Zachary Friggstad and Ramin Mousavi. A constant-factor approximation for quasi-bipartite directed steiner tree on minor-free graphs. arXiv preprint, 2021. URL:
  11. 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
  12. Fabrizio Grandoni, Bundit Laekhanukit, and Shi Li. O(log 2k/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
  13. 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
  14. Marek Karpinski and Alexander Zelikovsky. New approximation algorithms for the steiner tree problems. Journal of Combinatorial Optimization, 1(1):47-65, 1997. Google Scholar
  15. Richard J Lipton and Robert Endre Tarjan. A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36(2):177-189, 1979. Google Scholar
  16. Richard J Lipton and Robert Endre Tarjan. Applications of a planar separator theorem. SIAM journal on computing, 9(3):615-627, 1980. Google Scholar
  17. 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
  18. Neil Robertson and Paul D Seymour. Graph minors. xvi. excluding a non-planar graph. Journal of Combinatorial Theory, Series B, 89(1):43-76, 2003. Google Scholar
  19. Gabriel Robins and Alexander Zelikovsky. Tighter bounds for graph steiner tree approximation. SIAM Journal on Discrete Mathematics, 19(1):122-134, 2005. Google Scholar
  20. Mikkel Thorup. Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM (JACM), 51(6):993-1024, 2004. Google Scholar
  21. Alexander Zelikovsky. A series of approximation algorithms for the acyclic directed steiner tree problem. Algorithmica, 18(1):99-110, 1997. Google Scholar
  22. Alexander Z Zelikovsky. An 11/6-approximation algorithm for the network steiner problem. Algorithmica, 9(5):463-470, 1993. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail