Document Open Access Logo

From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs

Authors Chandra Chekuri, Rhea Jain, Shubhang Kulkarni , Da Wei Zheng , Weihao Zhu

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2024.42.pdf
  • Filesize: 0.84 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Discrete optimization
  • Theory of computation → Routing and network design problems
Keywords
  • Directed Planar Graphs
  • Submodular Functions
  • Steiner Tree
  • Network Design

Metrics

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

Abstract

In the Directed Steiner Tree (DST) problem the input is a directed edge-weighted graph G = (V,E), a root vertex r and a set S ⊆ V of k terminals. The goal is to find a min-cost subgraph that connects r to each of the terminals. DST admits an O(log² k/log log k)-approximation in quasi-polynomial time [Grandoni et al., 2022; Rohan Ghuge and Viswanath Nagarajan, 2022], and an O(k^{ε})-approximation for any fixed ε > 0 in polynomial-time [Alexander Zelikovsky, 1997; Moses Charikar et al., 1999]. Resolving the existence of a polynomial-time poly-logarithmic approximation is a major open problem in approximation algorithms. In a recent work, Friggstad and Mousavi [Zachary Friggstad and Ramin Mousavi, 2023] obtained a simple and elegant polynomial-time O(log k)-approximation for DST in planar digraphs via Thorup’s shortest path separator theorem [Thorup, 2004]. We build on their work and obtain several new results on DST and related problems.  
- We develop a tree embedding technique for rooted problems in planar digraphs via an interpretation of the recursion in [Zachary Friggstad and Ramin Mousavi, 2023]. Using this we obtain polynomial-time poly-logarithmic approximations for Group Steiner Tree [Naveen Garg et al., 2000], Covering Steiner Tree [Goran Konjevod et al., 2002] and the Polymatroid Steiner Tree [Gruia Călinescu and Alexander Zelikovsky, 2005] problems in planar digraphs. All these problems are hard to approximate to within a factor of Ω(log² n/log log n) even in trees [Eran Halperin and Robert Krauthgamer, 2003; Grandoni et al., 2022]. 
- We prove that the natural cut-based LP relaxation for DST has an integrality gap of O(log² k) in planar digraphs. This is in contrast to general graphs where the integrality gap of this LP is known to be Ω(√k) [Leonid Zosin and Samir Khuller, 2002] and Ω(n^{δ}) for some fixed δ > 0 [Shi Li and Bundit Laekhanukit, 2022]. 
- We combine the preceding results with density based arguments to obtain poly-logarithmic approximations for the multi-rooted versions of the problems in planar digraphs. For DST our result improves the O(R + log k) approximation of [Zachary Friggstad and Ramin Mousavi, 2023] when R = ω(log² k).

Cite As Get BibTex

Chandra Chekuri, Rhea Jain, Shubhang Kulkarni, Da Wei Zheng, and Weihao Zhu. From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 42:1-42:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.42

Author Details

Chandra Chekuri
  • University of Illinois, Urbana-Champaign, IL, USA
Rhea Jain
  • University of Illinois, Urbana-Champaign, IL, USA
Shubhang Kulkarni
  • University of Illinois, Urbana-Champaign, IL, USA
Da Wei Zheng
  • University of Illinois, Urbana-Champaign, IL, USA
Weihao Zhu
  • University of Illinois, Urbana-Champaign, IL, USA

Funding

Chandra Chekuri and Rhea Jain are partially supported by NSF grants CCF-1907937 and CCF-2402667. Weihao Zhu acknowledges funding support from a graduate fellowship of the CS department.

References

  1. Amit Agarwal and Moses Charikar. On the advantage of network coding for improving network throughput. In 2004 IEEE Information Theory Workshop, San Antonio, TX, USA, 24-29 October, 2004, pages 247-249. IEEE, 2004. URL: https://doi.org/10.1109/ITW.2004.1405308.
  2. MohammadHossein Bateni, Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dániel Marx. A PTAS for planar group steiner tree via spanner bootstrapping and prize collecting. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 570-583. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897549.
  3. Glencora Borradaile, Philip N. Klein, and Claire Mathieu. An O(n log n) approximation scheme for steiner tree in planar graphs. ACM Trans. Algorithms, 5(3):31:1-31:31, 2009. URL: https://doi.org/10.1145/1541885.1541892.
  4. Jaroslaw Byrka, Fabrizio Grandoni, Thomas Rothvoß, and Laura Sanità. Steiner tree approximation via iterative randomized rounding. J. ACM, 60(1):6:1-6:33, 2013. URL: https://doi.org/10.1145/2432622.2432628.
  5. Gruia Călinescu and Alexander Zelikovsky. The polymatroid steiner problems. J. Comb. Optim., 9(3):281-294, 2005. URL: https://doi.org/10.1007/S10878-005-1412-9.
  6. Parinya Chalermsook, Fabrizio Grandoni, and Bundit Laekhanukit. On survivable set connectivity. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 25-36. SIAM, 2015. URL: https://doi.org/10.1137/1.9781611973730.3.
  7. Moses Charikar, Chandra Chekuri, To-Yat Cheung, Zuo Dai, Ashish Goel, Sudipto Guha, and Ming Li. Approximation algorithms for directed steiner problems. J. Algorithms, 33(1):73-91, 1999. URL: https://doi.org/10.1006/JAGM.1999.1042.
  8. Moses Charikar, Chandra Chekuri, Ashish Goel, and Sudipto Guha. Rounding via trees: deterministic approximation algorithms for group steiner trees and k-median. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, STOC ’98, pages 114-123, New York, NY, USA, May 1998. Association for Computing Machinery. URL: https://doi.org/10.1145/276698.276719.
  9. Chandra Chekuri, Guy Even, Anupam Gupta, and Danny Segev. Set connectivity problems in undirected graphs and the directed steiner network problem. ACM Trans. Algorithms, 7(2):18:1-18:17, 2011. URL: https://doi.org/10.1145/1921659.1921664.
  10. Chandra Chekuri, Guy Even, and Guy Kortsarz. A greedy approximation algorithm for the group steiner problem. Discrete Applied Mathematics, 154(1):15-34, 2006. URL: https://doi.org/10.1016/j.dam.2005.07.010.
  11. Chandra Chekuri, Rhea Jain, Shubhang Kulkarni, Da Wei Zheng, and Weihao Zhu. From directed steiner tree to directed polymatroid steiner tree in planar graphs, 2024. Google Scholar
  12. Chandra Chekuri and Martin Pál. A recursive greedy algorithm for walks in directed graphs. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), 23-25 October 2005, Pittsburgh, PA, USA, Proceedings, pages 245-253. IEEE Computer Society, 2005. URL: https://doi.org/10.1109/SFCS.2005.9.
  13. Q. Chen, B. Laekhanukit, C. Liao, and Y. Zhang. Survivable network design revisited: Group-connectivity. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 278-289, Los Alamitos, CA, USA, November 2022. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS54457.2022.00033.
  14. Miroslav Chlebík and Janka Chlebíková. The steiner tree problem on graphs: Inapproximability results. Theor. Comput. Sci., 406(3):207-214, 2008. URL: https://doi.org/10.1016/J.TCS.2008.06.046.
  15. Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Philip N. Klein. Node-weighted steiner tree and group steiner tree in planar graphs. ACM Trans. Algorithms, 10(3):13:1-13:20, 2014. URL: https://doi.org/10.1145/2601070.
  16. Guy Even, Guy Kortsarz, and Wolfgang Slany. On network design problems: fixed cost flows and the covering steiner problem. ACM Trans. Algorithms, 1(1):74-101, 2005. URL: https://doi.org/10.1145/1077464.1077470.
  17. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci., 69(3):485-497, 2004. URL: https://doi.org/10.1016/J.JCSS.2004.04.011.
  18. Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  19. Christina Fragouli and Emina Soljanin. Network coding applications. Found. Trends Netw., 2(2):135-269, 2007. URL: https://doi.org/10.1561/1300000013.
  20. Christina Fragouli and Emina Soljanin. Network coding fundamentals. Found. Trends Netw., 2(1), 2007. URL: https://doi.org/10.1561/1300000003.
  21. András Frank. Connections in combinatorial optimization, volume 38. Oxford University Press Oxford, 2011. Google Scholar
  22. Zachary Friggstad, Jochen Könemann, Young Kun-Ko, Anand Louis, Mohammad Shadravan, and Madhur Tulsiani. Linear programming hierarchies suffice for directed steiner tree. In Jon Lee and Jens Vygen, editors, Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Bonn, Germany, June 23-25, 2014. Proceedings, volume 8494 of Lecture Notes in Computer Science, pages 285-296. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-07557-0_24.
  23. 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, June 22-24, 2016, Reykjavik, Iceland, volume 53 of LIPIcs, pages 3:1-3:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.SWAT.2016.3.
  24. Zachary Friggstad and Ramin Mousavi. A constant-factor approximation for quasi-bipartite directed steiner tree on minor-free graphs. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023, September 11-13, 2023, Atlanta, Georgia, USA, 275:13:1-13:18, 2023. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2023.13.
  25. 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, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 63:1-63:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.63.
  26. Naveen Garg, Goran Konjevod, and R. Ravi. A polylogarithmic approximation algorithm for the group steiner tree problem. J. Algorithms, 37(1):66-84, 2000. URL: https://doi.org/10.1006/JAGM.2000.1096.
  27. Rohan Ghuge and Viswanath Nagarajan. Quasi-polynomial algorithms for submodular tree orienteering and directed network design problems. Math. Oper. Res., 47(2):1612-1630, 2022. URL: https://doi.org/10.1287/MOOR.2021.1181.
  28. Michel X. Goemans, Neil Olver, Thomas Rothvoß, and Rico Zenklusen. Matroids and integrality gaps for hypergraphic steiner tree relaxations. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 1161-1176. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214081.
  29. 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. SIAM Journal on Computing, 52(2):298-322, 2022. URL: https://doi.org/10.1137/20M1312988.
  30. A. Gupta and J. Könemann. Approximation algorithms for network design: A survey. Surveys in Operations Research and Management Science, 16(1):3-20, 2011. Google Scholar
  31. Anupam Gupta and Aravind Srinivasan. An improved approximation ratio for the covering steiner problem. Theory Comput., 2(3):53-64, 2006. URL: https://doi.org/10.4086/TOC.2006.V002A003.
  32. Eran Halperin, Guy Kortsarz, Robert Krauthgamer, Aravind Srinivasan, and Nan Wang. Integrality ratio for group steiner trees and directed steiner trees. SIAM J. Comput., 36(5):1494-1511, 2007. URL: https://doi.org/10.1137/S0097539704445718.
  33. Eran Halperin and Robert Krauthgamer. Polylogarithmic inapproximability. In Lawrence L. Larmore and Michel X. Goemans, editors, Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pages 585-594. ACM, 2003. URL: https://doi.org/10.1145/780542.780628.
  34. Goran Konjevod, R. Ravi, and Aravind Srinivasan. Approximation algorithms for the covering steiner problem. Random Struct. Algorithms, 20(3):465-482, 2002. URL: https://doi.org/10.1002/RSA.10038.
  35. G. Kortsarz and Z. Nutov. Approximating minimum cost connectivity problems. In Dagstuhl Seminar Proceedings. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010. Google Scholar
  36. Shi Li and Bundit Laekhanukit. Polynomial integrality gap of flow LP for directed steiner tree. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 3230-3236. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.126.
  37. Harald Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In Cynthia Dwork, editor, Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 255-264. ACM, 2008. URL: https://doi.org/10.1145/1374376.1374415.
  38. Gabriele Reich and Peter Widmayer. Beyond steiner’s problem: A VLSI oriented generalization. In Manfred Nagl, editor, Graph-Theoretic Concepts in Computer Science, 15th International Workshop, WG '89, Castle Rolduc, The Netherlands, June 14-16, 1989, Proceedings, volume 411 of Lecture Notes in Computer Science, pages 196-210. Springer, 1989. URL: https://doi.org/10.1007/3-540-52292-1_14.
  39. Thomas Rothvoß. Directed steiner tree and the lasserre hierarchy. CoRR, abs/1111.5473, 2011. URL: https://arxiv.org/abs/1111.5473.
  40. Alexander Schrijver et al. Combinatorial optimization: polyhedra and efficiency. Springer, 2003. Google Scholar
  41. Mikkel Thorup. Compact oracles for reachability and approximate distances in planar digraphs. J. ACM, 51(6):993-1024, November 2004. URL: https://doi.org/10.1145/1039488.1039493.
  42. Vijay V Vazirani. Approximation algorithms, volume 1. Springer, 2001. Google Scholar
  43. David P Williamson and David B Shmoys. The design of approximation algorithms. Cambridge university press, 2011. Google Scholar
  44. Laurence A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Comb., 2(4):385-393, 1982. URL: https://doi.org/10.1007/BF02579435.
  45. Alexander Zelikovsky. A series of approximation algorithms for the acyclic directed steiner tree problem. Algorithmica, 18(1):99-110, 1997. URL: https://doi.org/10.1007/BF02523690.
  46. Leonid Zosin and Samir Khuller. On directed steiner trees. In David Eppstein, editor, Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 6-8, 2002, San Francisco, CA, USA, pages 59-63. ACM/SIAM, 2002. URL: http://dl.acm.org/citation.cfm?id=545381.545388.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

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