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Steiner Tree Parameterized by Multiway Cut and Even Less

Authors Bart M.P. Jansen , Céline M.F. Swennenhuis

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Fixed parameter tractability
Keywords
  • fixed-parameter tractability
  • Steiner Tree
  • structural parameterization
  • H-treewidth

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Abstract

In the Steiner Tree problem we are given an undirected edge-weighted graph as input, along with a set K of vertices called terminals. The task is to output a minimum-weight connected subgraph that spans all the terminals. The famous Dreyfus-Wagner algorithm running in 3^{|K|}poly(n) time shows that the problem is fixed-parameter tractable parameterized by the number of terminals. We present fixed-parameter tractable algorithms for Steiner Tree using structurally smaller parameterizations.
Our first result concerns the parameterization by a multiway cut S of the terminals, which is a vertex set S (possibly containing terminals) such that each connected component of G-S contains at most one terminal. We show that Steiner Tree can be solved in 2^{𝒪(|S|log|S|)}poly(n) time and polynomial space, where S is a minimum multiway cut for K. The algorithm is based on the insight that, after guessing how an optimal Steiner tree interacts with a multiway cut S, computing a minimum-cost solution of this type can be formulated as minimum-cost bipartite matching.
Our second result concerns a new hybrid parameterization called K-free treewidth that simultaneously refines the number of terminals |K| and the treewidth of the input graph. By utilizing recent work on ℋ-Treewidth in order to find a corresponding decomposition of the graph, we give an algorithm that solves Steiner Tree in time 2^{𝒪(k)} poly(n), where k denotes the K-free treewidth of the input graph. To obtain this running time, we show how the rank-based approach for solving Steiner Tree parameterized by treewidth can be extended to work in the setting of K-free treewidth, by exploiting existing algorithms parameterized by |K| to compute the table entries of leaf bags of a tree K-free decomposition.

Cite As Get BibTex

Bart M.P. Jansen and Céline M.F. Swennenhuis. Steiner Tree Parameterized by Multiway Cut and Even Less. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 76:1-76:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.76

Author Details

Bart M.P. Jansen
  • Eindhoven University of Technology, The Netherlands
Céline M.F. Swennenhuis
  • Eindhoven University of Technology, The Netherlands

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 803421, ReduceSearch).

References

  1. Akanksha Agrawal, Lawqueen Kanesh, Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Deleting, eliminating and decomposing to hereditary classes are all FPT-equivalent. In Proceedings of SODA 2022, pages 1976-2004, 2022. URL: https://doi.org/10.1137/1.9781611977073.79.
  2. Akanksha Agrawal and M. S. Ramanujan. Distance from triviality 2.0: Hybrid parameterizations. In Proc. 33rd IWOCA, volume 13270 of Lecture Notes in Computer Science, pages 3-20. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-06678-8_1.
  3. Andreas Björklund, Thore Husfeldt, and Nina Taslaman. Shortest cycle through specified elements. In Proc. 23rd SODA, pages 1747-1753. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.139.
  4. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Information and Computation, 243:86-111, 2015. URL: https://doi.org/10.1016/j.ic.2014.12.008.
  5. Jannis Bulian and Anuj Dawar. Graph isomorphism parameterized by elimination distance to bounded degree. Algorithmica, 75(2):363-382, 2016. URL: https://doi.org/10.1007/S00453-015-0045-3.
  6. Jianer Chen, Yang Liu, and Songjian Lu. An improved parameterized algorithm for the minimum node multiway cut problem. Algorithmica, 55(1):1-13, 2009. URL: https://doi.org/10.1007/S00453-007-9130-6.
  7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  8. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michał Pilipczuk, Johan M.M. Van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. ACM Transactions on Algorithms (TALG), 18(2):1-31, 2022. URL: https://doi.org/10.1145/3506707.
  9. 11th DIMACS Implementation Challenge, 2014. Accessed 21 Nov 2023. URL: https://dimacs11.zib.de/.
  10. 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.
  11. Eduard Eiben, Robert Ganian, Thekla Hamm, and O joung Kwon. Measuring what matters: A hybrid approach to dynamic programming with treewidth. Journal of Computer and System Sciences, 121:57-75, 2021. URL: https://doi.org/10.1016/j.jcss.2021.04.005.
  12. Fedor V Fomin, Petteri Kaski, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Parameterized single-exponential time polynomial space algorithm for Steiner tree. SIAM Journal on Discrete Mathematics, 33(1):327-345, 2019. URL: https://doi.org/10.1137/17M1140030.
  13. Bernhard Fuchs, Walter Kern, D Molle, Stefan Richter, Peter Rossmanith, and Xinhui Wang. Dynamic programming for minimum Steiner trees. Theory of Computing Systems, 41:493-500, 2007. URL: https://doi.org/10.1007/S00224-007-1324-4.
  14. Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: Distance from triviality. In Proc. 1st IWPEC, pages 162-173. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-28639-4_15.
  15. Trey Ideker, Owen Ozier, Benno Schwikowski, and Andrew F Siegel. Discovering regulatory and signalling circuits in molecular interaction networks. Bioinformatics, 18:S233-S240, 2002. URL: https://doi.org/10.1093/bioinformatics/18.suppl_1.S233.
  16. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/JCSS.2000.1727.
  17. Bart M. P. Jansen, Jari J. H. de Kroon, and Michał Włodarczyk. Vertex deletion parameterized by elimination distance and even less. In Proc. 53rd STOC, pages 1757-1769. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451068.
  18. Bart M. P. Jansen, Jari J. H. de Kroon, and Michal Wlodarczyk. 5-approximation for ℋ-treewidth essentially as fast as ℋ-deletion parameterized by solution size. In Proc. 31st ESA, volume 274 of LIPIcs, pages 66:1-66:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.66.
  19. Richard M. Karp. Reducibility among combinatorial problems. In Proceedings of a symposium on the Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  20. Ton Kloks. Treewidth: computations and approximations, volume 842. Springer Science & Business Media, 1994. URL: https://doi.org/10.1007/BFb0045375.
  21. Thomas Lengauer. Combinatorial algorithms for integrated circuit layout. XApplicable Theory in Computer Science. Springer Science & Business Media, 2012. URL: https://doi.org/10.1007/978-3-322-92106-2.
  22. Ivana Ljubić. Solving Steiner trees: Recent advances, challenges, and perspectives. Networks, 77(2):177-204, 2021. URL: https://doi.org/10.1002/net.22005.
  23. Daniel Lokshtanov and Jesper Nederlof. Saving space by algebraization. In Proc. 42nd STOC, pages 321-330. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806735.
  24. Jesper Nederlof. Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica, 65(4):868-884, 2013. URL: https://doi.org/10.1007/S00453-012-9630-X.
  25. Jaroslav Nešetřil and Patrice Ossona de Mendez. Chapter 6: Bounded height trees and tree-depth. In Sparsity: Graphs, Structures, and Algorithms, volume 28 of Algorithms and Combinatorics, pages 115-144. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
  26. PACE 2018, 2018. Accessed 21 Nov 2023. URL: https://pacechallenge.org/2018/.
  27. Mauricio G.C. Resende and Panos M. Pardalos. Handbook of optimization in telecommunications. Springer Science & Business Media, 2008. URL: https://doi.org/10.1007/978-0-387-30165-5.
  28. Olga Russakovsky and Andrew Y. Ng. A Steiner tree approach to efficient object detection. In Proc. 23rd CVPR, pages 1070-1077. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/CVPR.2010.5540097.
  29. Tom van Roozendaal. Fixed-parameter tractable algorithms for refined parameterizations of graph problems. Master’s thesis, Eindhoven University of Technology, 2023. URL: https://research.tue.nl/en/studentTheses/fixed-parameter-tractable-algorithms-for-refined-parameterization.
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