Document Open Access Logo

Subexponential Parameterized Directed Steiner Network Problems on Planar Graphs: A Complete Classification

Authors Esther Galby , Sándor Kisfaludi-Bak , Dániel Marx , Roohani Sharma

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.67.pdf
  • Filesize: 0.93 MB
  • 19 pages

Document Identifiers

Author Details

Esther Galby
  • Chalmers University of Technology, Gothenburg, Sweden
Sándor Kisfaludi-Bak
  • Department of Computer Science, Aalto University, Finland
Dániel Marx
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Roohani Sharma
  • Department of Informatics, University of Bergen, Norway

Cite AsGet BibTex

Esther Galby, Sándor Kisfaludi-Bak, Dániel Marx, and Roohani Sharma. Subexponential Parameterized Directed Steiner Network Problems on Planar Graphs: A Complete Classification. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.67

Abstract

In the Directed Steiner Network problem, the input is a directed graph G, a set T ⊆ V(G) of k terminals, and a demand graph D on T. The task is to find a subgraph H ⊆ G with the minimum number of edges such that for every (s,t) ∈ E(D), the solution H contains a directed s → t path. The goal of this paper is to investigate how the complexity of the problem depends on the demand pattern in planar graphs. Formally, if 𝒟 is a class of directed graphs, then the 𝒟-Steiner Network (𝒟-DSN) problem is the special case where the demand graph D is restricted to be from 𝒟. We give a complete characterization of the behavior of every 𝒟-DSN problem on planar graphs. We classify every class 𝒟 closed under transitive equivalence and identification of vertices into three cases: assuming ETH, either the problem is 1) solvable in time 2^O(k)⋅n^O(1), i.e., FPT parameterized by the number k of terminals, but not solvable in time 2^o(k)⋅n^O(1), 2) solvable in time f(k)⋅n^O(√k), but cannot be solved in time f(k)⋅n^o(√k), or 3) solvable in time f(k)⋅n^O(k), but cannot be solved in time f(k)⋅n^o(k). Our result is a far-reaching generalization and unification of earlier results on Directed Steiner Tree, Directed Steiner Network, and Strongly Connected Steiner Subgraph on planar graphs. As an important step of our lower bound proof, we discover a rare example of a genuinely planar problem (i.e., described by a planar graph and two sets of vertices) that cannot be solved in time f(k)⋅n^o(k): given two sets of terminals S and T with |S|+|T| = k, find a subgraph with minimum number of edges such that every vertex of T is reachable from every vertex of S.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Graph algorithms analysis
Keywords
  • Directed Steiner Network
  • Sub-exponential algorithm

Metrics

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

Related Versions

References

  1. MohammadHossein Bateni, Chandra Chekuri, Alina Ene, Mohammad Taghi Hajiaghayi, Nitish Korula, and Dániel Marx. Prize-collecting Steiner Problems on Planar Graphs. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 1028-1049, 2011. URL: https://doi.org/10.1137/1.9781611973082.79.
  2. MohammadHossein Bateni, Mohammad Taghi Hajiaghayi, and Dániel Marx. Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth. J. ACM, 58(5):21:1-21:37, 2011. URL: https://doi.org/10.1145/2027216.2027219.
  3. Piotr Berman, Arnab Bhattacharyya, Konstantin Makarychev, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Approximation algorithms for spanner problems and directed steiner forest. Inf. Comput., 222:93-107, 2013. URL: https://doi.org/10.1016/j.ic.2012.10.007.
  4. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets möbius: fast subset convolution. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 67-74. ACM, 2007. URL: https://doi.org/10.1145/1250790.1250801.
  5. 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.
  6. Rajesh Chitnis, Hossein Esfandiari, Mohammad Taghi Hajiaghayi, Rohit Khandekar, Guy Kortsarz, and Saeed Seddighin. A Tight Algorithm for Strongly Connected Steiner Subgraph on Two Terminals with Demands. Algorithmica, 77(4):1216-1239, 2017. URL: https://doi.org/10.1007/s00453-016-0145-8.
  7. Rajesh Chitnis, Andreas Emil Feldmann, and Pasin Manurangsi. Parameterized Approximation Algorithms for Bidirected Steiner Network Problems. In 26th Annual European Symposium on Algorithms, ESA 2018, August 20-22, 2018, Helsinki, Finland, pages 20:1-20:16, 2018. URL: https://doi.org/10.4230/LIPIcs.ESA.2018.20.
  8. Rajesh Hemant Chitnis, Andreas Emil Feldmann, Mohammad Taghi Hajiaghayi, and Dániel Marx. Tight bounds for planar strongly connected steiner subgraph with fixed number of terminals (and extensions). SIAM J. Comput., 49(2):318-364, 2020. URL: https://doi.org/10.1137/18M122371X.
  9. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer Publishing Company, Incorporated, 1st edition, 2015. Google Scholar
  10. Éric Colin de Verdière. Multicuts in planar and bounded-genus graphs with bounded number of terminals. Algorithmica, 78(4):1206-1224, 2017. URL: https://doi.org/10.1007/s00453-016-0258-0.
  11. Pavel Dvorák, Andreas Emil Feldmann, Dusan Knop, Tomás Masarík, Tomas Toufar, and Pavel Veselý. Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices. In 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France, pages 26:1-26:15, 2018. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.26.
  12. Eduard Eiben, Dusan Knop, Fahad Panolan, and Ondrej Suchý. Complexity of the Steiner Network Problem with Respect to the Number of Terminals. In Rolf Niedermeier and Christophe Paul, editors, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019), volume 126 of Leibniz International Proceedings in Informatics (LIPIcs), pages 25:1-25:17, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2019.25.
  13. Jon Feldman and Matthias Ruhl. The Directed Steiner Network Problem is Tractable for a Constant Number of Terminals. SIAM J. Comput., 36(2):543-561, 2006. URL: https://doi.org/10.1137/S0097539704441241.
  14. Andreas Emil Feldmann and Dániel Marx. The complexity landscape of fixed-parameter directed steiner network problems. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, volume 55 of LIPIcs, pages 27:1-27:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.27.
  15. Fedor V. Fomin, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Subexponential parameterized algorithms for planar and apex-minor-free graphs via low treewidth pattern covering. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 515-524. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.62.
  16. Esther Galby, Sándor Kisfaludi-Bak, Dániel Marx, and Roohani Sharma. Subexponential parameterized directed steiner network problems on planar graphs: a complete classification, 2022. URL: https://arxiv.org/abs/2208.06015.
  17. Jiong Guo, Rolf Niedermeier, and Ondrej Suchý. Parameterized complexity of arc-weighted directed steiner problems. SIAM J. Discrete Math., 25(2):583-599, 2011. URL: https://doi.org/10.1137/100794560.
  18. S. Louis Hakimi. Steiner’s problem in graphs and its implications. Networks, 1(2):113-133, 1971. URL: https://doi.org/10.1002/net.3230010203.
  19. Richard M. Karp. Reducibility Among Combinatorial Problems. In Proceedings of a Symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, pages 85-103, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  20. Philip N. Klein and Dániel Marx. Solving planar k-terminal cut in O(n^{c√k}) time. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, volume 7391 of Lecture Notes in Computer Science, pages 569-580. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_48.
  21. Philip N. Klein and Dániel Marx. A subexponential parameterized algorithm for subset TSP on planar graphs. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1812-1830. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.131.
  22. A Levin. Algorithm for the shortest connection of a group of graph vertices. Soviet Math. Dokl., 12:1477-1481, 1971. URL: https://doi.org/10.4086/toc.2010.v006a005.
  23. Chung-Lun Li, S. Thomas McCormick, and David Simchi-Levi. The point-to-point delivery and connection problems: complexity and algorithms. Discrete Applied Mathematics, 36(3):267-292, 1992. URL: https://doi.org/10.1016/0166-218X(92)90258-C.
  24. Daniel Lokshtanov, Saket Saurabh, and Magnus Wahlström. Subexponential parameterized odd cycle transversal on planar graphs. In Deepak D'Souza, Telikepalli Kavitha, and Jaikumar Radhakrishnan, editors, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012, December 15-17, 2012, Hyderabad, India, volume 18 of LIPIcs, pages 424-434. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2012.424.
  25. Dániel Marx. On the Optimality of Planar and Geometric Approximation Schemes. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), October 20-23, 2007, Providence, RI, USA, Proceedings, pages 338-348, 2007. URL: https://doi.org/10.1109/FOCS.2007.50.
  26. Dániel Marx. A tight lower bound for planar multiway cut with fixed number of terminals. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, volume 7391 of Lecture Notes in Computer Science, pages 677-688. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_57.
  27. Dániel Marx, Marcin Pilipczuk, and Michal Pilipczuk. On subexponential parameterized algorithms for steiner tree and directed subset TSP on planar graphs. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 474-484. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00052.
  28. Dániel Marx and Michal Pilipczuk. Optimal parameterized algorithms for planar facility location problems using voronoi diagrams. ACM Trans. Algorithms, 18(2):13:1-13:64, 2022. URL: https://doi.org/10.1145/3483425.
  29. Madan Natu and Shu-Cherng Fang. The Point-to-point Connection Problem - Analysis and Algorithms. Discrete Applied Mathematics, 78(1-3):207-226, 1997. URL: https://doi.org/10.1016/S0166-218X(97)00010-3.
  30. Jesper Nederlof. Detecting and counting small patterns in planar graphs in subexponential parameterized time. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 1293-1306. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384261.
  31. S. Ramanathan. Multicast tree generation in networks with asymmetric links. IEEE/ACM Trans. Netw., 4(4):558-568, 1996. URL: https://doi.org/10.1109/90.532865.
  32. Hussein F. Salama, Douglas S. Reeves, and Yannis Viniotis. Evaluation of Multicast Routing Algorithms for Real-Time Communication on High-Speed Networks. IEEE Journal on Selected Areas in Communications, 15(3):332-345, 1997. URL: https://doi.org/10.1109/49.564132.
  33. Pawel Winter. Steiner problem in networks: A survey. Networks, 17(2):129-167, 1987. URL: https://doi.org/10.1002/net.3230170203.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact