Parameterized Algorithms for Steiner Forest in Bounded Width Graphs

Authors Andreas Emil Feldmann , Michael Lampis



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Andreas Emil Feldmann
  • Department of Computer Science, University of Sheffield, UK
Michael Lampis
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France

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Andreas Emil Feldmann and Michael Lampis. Parameterized Algorithms for Steiner Forest in Bounded Width Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 61:1-61:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.61

Abstract

In this paper we reassess the parameterized complexity and approximability of the well-studied Steiner Forest problem in several graph classes of bounded width. The problem takes an edge-weighted graph and pairs of vertices as input, and the aim is to find a minimum cost subgraph in which each given vertex pair lies in the same connected component. It is known that this problem is APX-hard in general, and NP-hard on graphs of treewidth 3, treedepth 4, and feedback vertex set size 2. However, Bateni, Hajiaghayi and Marx [JACM, 2011] gave an approximation scheme with a runtime of n^O(k²/ε) on graphs of treewidth k. Our main result is a much faster efficient parameterized approximation scheme (EPAS) with a runtime of 2^O(k²/ε log k/ε)⋅n^O(1). If k instead is the vertex cover number of the input graph, we show how to compute the optimum solution in 2^O(k log k)⋅n^O(1) time, and we also prove that this runtime dependence on k is asymptotically best possible, under ETH. Furthermore, if k is the size of a feedback edge set, then we obtain a faster 2^O(k)⋅n^O(1) time algorithm, which again cannot be improved under ETH.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Steiner Forest
  • Approximation Algorithms
  • FPT algorithms

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References

  1. Ajit Agrawal, Philip Klein, and Ramamoorthi Ravi. When trees collide: An approximation algorithm for the generalized steiner problem on networks. In Proceedings of the twenty-third annual ACM symposium on Theory of computing, pages 134-144, 1991. Google Scholar
  2. Brenda S Baker. Approximation algorithms for np-complete problems on planar graphs. Journal of the ACM (JACM), 41(1):153-180, 1994. Google Scholar
  3. Yair Bartal and Lee-Ad Gottlieb. Near-linear time approximation schemes for steiner tree and forest in low-dimensional spaces. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1028-1041, 2021. Google Scholar
  4. 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
  5. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets möbius: fast subset convolution. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 67-74, 2007. Google Scholar
  6. 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. Google Scholar
  7. Hans L. Bodlaender and Torben Hagerup. Parallel algorithms with optimal speedup for bounded treewidth. SIAM J. Comput., 27(6):1725-1746, 1998. URL: https://doi.org/10.1137/S0097539795289859.
  8. Hans L Bodlaender, Matthew Johnson, Barnaby Martin, Jelle J Oostveen, Sukanya Pandey, Daniel Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity framework for forbidden subgraphs iv: The steiner forest problem. arXiv preprint, 2023. URL: https://arxiv.org/abs/2305.01613.
  9. 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
  10. 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
  11. Xiuzhen Cheng and Ding-Zhu Du. Steiner trees in industry, volume 11. Springer Science & Business Media, 2013. Google Scholar
  12. Rajesh Chitnis, Andreas Emil Feldmann, and Pasin Manurangsi. Parameterized approximation algorithms for bidirected steiner network problems. ACM Transactions on Algorithms (TALG), 17(2):1-68, 2021. Google Scholar
  13. Miroslav Chlebík and Janka Chlebíková. The steiner tree problem on graphs: Inapproximability results. Theoretical Computer Science, 406(3):207-214, 2008. Google Scholar
  14. 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.
  15. Stuart E Dreyfus and Robert A Wagner. The steiner problem in graphs. Networks, 1(3):195-207, 1971. Google Scholar
  16. Ding-Zhu Du, JM Smith, and J Hyam Rubinstein. Advances in Steiner trees, volume 6. Springer Science & Business Media, 2013. Google Scholar
  17. Pavel Dvorák, Andreas E Feldmann, Dusan Knop, Tomás Masarík, Tomas Toufar, and Pavel Vesely. Parameterized approximation schemes for steiner trees with small number of steiner vertices. SIAM Journal on Discrete Mathematics, 35(1):546-574, 2021. Google Scholar
  18. David Eisenstat, Philip Klein, and Claire Mathieu. An efficient polynomial-time approximation scheme for steiner forest in planar graphs. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 626-638. SIAM, 2012. Google Scholar
  19. Andreas Emil Feldmann, Euiwoong Lee, and Pasin Manurangsi. A survey on approximation in parameterized complexity: Hardness and algorithms. Algorithms, 13(6):146, 2020. Google Scholar
  20. Bernhard Fuchs, Walter Kern, D Molle, Stefan Richter, Peter Rossmanith, and Xinhui Wang. Dynamic programming for minimum steiner trees. Theory of Computing Systems, 41(3):493-500, 2007. Google Scholar
  21. Elisabeth Gassner. The steiner forest problem revisited. Journal of Discrete Algorithms, 8(2):154-163, 2010. Google Scholar
  22. Tatsuya Gima, Tesshu Hanaka, Masashi Kiyomi, Yasuaki Kobayashi, and Yota Otachi. Exploring the gap between treedepth and vertex cover through vertex integrity. Theor. Comput. Sci., 918:60-76, 2022. URL: https://doi.org/10.1016/J.TCS.2022.03.021.
  23. Anupam Gupta and Jochen Könemann. Approximation algorithms for network design: A survey. Surveys in Operations Research and Management Science, 16(1):3-20, 2011. Google Scholar
  24. Frank K Hwang and Dana S Richards. Steiner tree problems. Networks, 22(1):55-89, 1992. Google Scholar
  25. Richard M Karp. On the computational complexity of combinatorial problems. Networks, 5(1):45-68, 1975. Google Scholar
  26. Michael Lampis, Nikolaos Melissinos, and Manolis Vasilakis. Parameterized max min feedback vertex set. In Jérôme Leroux, Sylvain Lombardy, and David Peleg, editors, 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023, August 28 to September 1, 2023, Bordeaux, France, volume 272 of LIPIcs, pages 62:1-62:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.MFCS.2023.62.
  27. Michael Lampis and Manolis Vasilakis. Structural parameterizations for two bounded degree problems revisited. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 77:1-77:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.77.
  28. Ivana Ljubic. Solving steiner trees: Recent advances, challenges, and perspectives. Networks, 77(2):177-204, 2021. Google Scholar
  29. Jesper Nederlof. Fast polynomial-space algorithms using möbius inversion: Improving on steiner tree and related problems. In International Colloquium on Automata, Languages, and Programming, pages 713-725. Springer, 2009. Google Scholar
  30. Satish B Rao and Warren D Smith. Approximating geometrical graphs via “spanners” and “banyans”. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 540-550, 1998. Google Scholar
  31. R Ravi. A primal-dual approximation algorithm for the steiner forest problem. Information processing letters, 50(4):185-189, 1994. Google Scholar
  32. Hao Tang, Genggeng Liu, Xiaohua Chen, and Naixue Xiong. A survey on steiner tree construction and global routing for vlsi design. IEEE Access, 8:68593-68622, 2020. Google Scholar
  33. Stefan Voß. Steiner tree problems in telecommunications. In Handbook of optimization in telecommunications, pages 459-492. Springer, 2006. Google Scholar
  34. David P Williamson and David B Shmoys. The design of approximation algorithms. Cambridge university press, 2011. Google Scholar
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