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Cuts in Graphs with Matroid Constraints

Authors Aritra Banik, Fedor V. Fomin , Petr A. Golovach , Tanmay Inamdar , Satyabrata Jana , Saket Saurabh

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Author Details

Aritra Banik
  • National Institute of Science, Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, Odisha, India
Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Tanmay Inamdar
  • Indian Institute of Technology Jodhpur, India
Satyabrata Jana
  • University of Warwick, UK
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway

Funding

  • Fomin, Fedor V.: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Golovach, Petr A.: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Jana, Satyabrata: Supported by the Engineering and Physical Sciences Research Council (EPSRC) via the project MULTIPROCESS (grant no. EP/V044621/1).
  • Saurabh, Saket: The author is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416); and he also acknowledges the support of Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.

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Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, and Saket Saurabh. Cuts in Graphs with Matroid Constraints. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.17

Abstract

Vertex (s, t)-Cut and Vertex Multiway Cut are two fundamental graph separation problems in algorithmic graph theory. We study matroidal generalizations of these problems, where in addition to the usual input, we are given a representation R ∈ 𝔽^{r × n} of a linear matroid ℳ = (V(G), ℐ) of rank r in the input, and the goal is to determine whether there exists a vertex subset S ⊆ V(G) that has the required cut properties, as well as is independent in the matroid ℳ. We refer to these problems as Independent Vertex (s, t){-cut}, and Independent Multiway Cut, respectively. We show that these problems are fixed-parameter tractable (FPT) when parameterized by the solution size (which can be assumed to be equal to the rank of the matroid ℳ). These results are obtained by exploiting the recent technique of flow augmentation [Kim et al. STOC '22], combined with a dynamic programming algorithm on flow-paths á la [Feige and Mahdian, STOC '06] that maintains a representative family of solutions w.r.t. the given matroid [Marx, TCS '06; Fomin et al., JACM]. As a corollary, we also obtain FPT algorithms for the independent version of Odd Cycle Transversal. Further, our results can be generalized to other variants of the problems, e.g., weighted versions, or edge-deletion versions.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Matroids and greedoids
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • s-t-cut
  • multiway Cut
  • matroid
  • odd cycle transversal
  • feedback vertex set
  • fixed-parameter tractability

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References

  1. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, 1995. URL: https://doi.org/10.1145/210332.210337.
  2. Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, and Saket Saurabh. Cuts in graphs with matroid constraints, 2024. URL: https://arxiv.org/abs/2406.19134.
  3. Gruia Călinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput., 40(6):1740-1766, 2011. URL: https://doi.org/10.1137/080733991.
  4. M. Cygan, F. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  5. 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.
  6. Marek Cygan, Fabrizio Grandoni, and Danny Hermelin. Tight kernel bounds for problems on graphs with small degeneracy. ACM Trans. Algorithms, 13(3):43:1-43:22, 2017. URL: https://doi.org/10.1145/3108239.
  7. Uriel Feige and Mohammad Mahdian. Finding small balanced separators. In Jon M. Kleinberg, editor, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 375-384. ACM, 2006. URL: https://doi.org/10.1145/1132516.1132573.
  8. F. V. Fomin, D. Lokshtanov, and S. Saurabh. Efficient computation of representative sets with applications in parameterized and exact algorithms. In proceedings of SODA, pages 142-151, 2014. Google Scholar
  9. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM, 63(4):29:1-29:60, 2016. URL: https://doi.org/10.1145/2886094.
  10. Gagan Goel, Chinmay Karande, Pushkar Tripathi, and Lei Wang. Approximability of combinatorial problems with multi-agent submodular cost functions. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 755-764. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/FOCS.2009.81.
  11. MohammadTaghi Hajiaghayi, Rohit Khandekar, and Guy Kortsarz. Local search algorithms for the red-blue median problem. Algorithmica, 63(4):795-814, 2012. URL: https://doi.org/10.1007/S00453-011-9547-9.
  12. Zhong Huang and Xueliang Li. Hardness results for rainbow disconnection of graphs. CoRR, abs/1811.11939, 2018. URL: https://arxiv.org/abs/1811.11939.
  13. Stefanie Jegelka and Jeff Bilmes. Notes on graph cuts with submodular edge weights. In NIPS 2009 Workshop on Discrete Optimization in Machine Learning: Submodularity, Sparsity Polyhedra (DISCML), pages 1-6, 2009. Google Scholar
  14. Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. Flow-augmentation i: Directed graphs. arXiv preprint arXiv:2111.03450, 2021. Google Scholar
  15. Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. Solving hard cut problems via flow-augmentation. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 149-168. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.11.
  16. Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. Directed flow-augmentation. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 938-947. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520018.
  17. Ravishankar Krishnaswamy, Shi Li, and Sai Sandeep. Constant approximation for k-median and k-means with outliers via iterative rounding. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 646-659. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188882.
  18. Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, and Saket Saurabh. Deterministic truncation of linear matroids. ACM Trans. Algorithms, 14(2):14:1-14:20, 2018. URL: https://doi.org/10.1145/3170444.
  19. László Lovász. Flats in matroids and geometric graphs. In Combinatorial Surveys (Proc. 6th British Combinatorial Conference, pages 45-86, 1977. Google Scholar
  20. László Lovász. Graphs and geometry, volume 65 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2019. URL: https://doi.org/10.1090/coll/065.
  21. Pasin Manurangsi, Erel Segal-Halevi, and Warut Suksompong. On maximum bipartite matching with separation. Inf. Process. Lett., 182:106388, 2023. URL: https://doi.org/10.1016/J.IPL.2023.106388.
  22. Dániel Marx. Parameterized graph separation problems. Theor. Comput. Sci., 351(3):394-406, 2006. URL: https://doi.org/10.1016/J.TCS.2005.10.007.
  23. Dániel Marx. A parameterized view on matroid optimization problems. Theor. Comput. Sci., 410(44):4471-4479, 2009. URL: https://doi.org/10.1016/j.tcs.2009.07.027.
  24. Dániel Marx, Barry O'Sullivan, and Igor Razgon. Treewidth reduction for constrained separation and bipartization problems. In Jean-Yves Marion and Thomas Schwentick, editors, 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010, March 4-6, 2010, Nancy, France, volume 5 of LIPIcs, pages 561-572. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010. URL: https://doi.org/10.4230/LIPICS.STACS.2010.2485.
  25. Dániel Marx, Barry O'Sullivan, and Igor Razgon. Finding small separators in linear time via treewidth reduction. ACM Trans. Algorithms, 9(4):30:1-30:35, 2013. URL: https://doi.org/10.1145/2500119.
  26. Syed Mohammad Meesum, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Rank vertex cover as a natural problem for algebraic compression. SIAM J. Discret. Math., 33(3):1277-1296, 2019. URL: https://doi.org/10.1137/17M1154370.
  27. Rolf Niedermeier and Peter Rossmanith. On efficient fixed-parameter algorithms for weighted vertex cover. J. Algorithms, 47(2):63-77, 2003. URL: https://doi.org/10.1016/S0196-6774(03)00005-1.
  28. Bruce A. Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Oper. Res. Lett., 32(4):299-301, 2004. URL: https://doi.org/10.1016/J.ORL.2003.10.009.
  29. Hadas Shachnai and Meirav Zehavi. A multivariate framework for weighted FPT algorithms. J. Comput. Syst. Sci., 89:157-189, 2017. URL: https://doi.org/10.1016/J.JCSS.2017.05.003.
  30. Laurence A Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2(4):385-393, 1982. Google Scholar
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