Parameterized Max Min Feedback Vertex Set

Authors Michael Lampis , Nikolaos Melissinos , Manolis Vasilakis

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Author Details

Michael Lampis
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France
Nikolaos Melissinos
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Czech Republic
Manolis Vasilakis
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France


Work primarily conducted while Nikolaos Melissinos was affiliated with Université Paris-Dauphine.

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Michael Lampis, Nikolaos Melissinos, and Manolis Vasilakis. Parameterized Max Min Feedback Vertex Set. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Given a graph G and an integer k, Max Min FVS asks whether there exists a minimal set of vertices of size at least k whose deletion destroys all cycles. We present several results that improve upon the state of the art of the parameterized complexity of this problem with respect to both structural and natural parameters. Using standard DP techniques, we first present an algorithm of time tw^O(tw) n^O(1), significantly generalizing a recent algorithm of Gaikwad et al. of time vc^O(vc) n^O(1), where tw, vc denote the input graph’s treewidth and vertex cover respectively. Subsequently, we show that both of these algorithms are essentially optimal, since a vc^o(vc) n^O(1) algorithm would refute the ETH. With respect to the natural parameter k, the aforementioned recent work by Gaikwad et al. claimed an FPT branching algorithm with complexity 10^k n^O(1). We point out that this algorithm is incorrect and present a branching algorithm of complexity 9.34^k n^O(1).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • ETH
  • Feedback vertex set
  • Parameterized algorithms
  • Treewidth


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  1. Hassan AbouEisha, Shahid Hussain, Vadim V. Lozin, Jérôme Monnot, Bernard Ries, and Viktor Zamaraev. Upper domination: Towards a dichotomy through boundary properties. Algorithmica, 80(10):2799-2817, 2018. URL:
  2. Júlio Araújo, Marin Bougeret, Victor A. Campos, and Ignasi Sau. Introducing lop-kernels: A framework for kernelization lower bounds. Algorithmica, 84(11):3365-3406, 2022. URL:
  3. Júlio Araújo, Marin Bougeret, Victor A. Campos, and Ignasi Sau. Parameterized complexity of computing maximum minimal blocking and hitting sets. Algorithmica, 85(2):444-491, 2023. URL:
  4. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary. In SODA, pages 951-970. SIAM, 2020. URL:
  5. Cristina Bazgan, Ljiljana Brankovic, Katrin Casel, Henning Fernau, Klaus Jansen, Kim-Manuel Klein, Michael Lampis, Mathieu Liedloff, Jérôme Monnot, and Vangelis Th. Paschos. The many facets of upper domination. Theor. Comput. Sci., 717:2-25, 2018. URL:
  6. Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Yota Otachi. Grundy distinguishes treewidth from pathwidth. SIAM Journal on Discrete Mathematics, 36(3):1761-1787, 2022. URL:
  7. Benjamin Bergougnoux, Édouard Bonnet, Nick Brettell, and O-joung Kwon. Close relatives of feedback vertex set without single-exponential algorithms parameterized by treewidth. In IPEC, volume 180 of LIPIcs, pages 3:1-3:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  8. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015. URL:
  9. Marthe Bonamy, Lukasz Kowalik, Jesper Nederlof, Michal Pilipczuk, Arkadiusz Socala, and Marcin Wrochna. On directed feedback vertex set parameterized by treewidth. In WG, volume 11159 of Lecture Notes in Computer Science, pages 65-78. Springer, 2018. URL:
  10. Édouard Bonnet, Nick Brettell, O-joung Kwon, and Dániel Marx. Generalized feedback vertex set problems on bounded-treewidth graphs: Chordality is the key to single-exponential parameterized algorithms. Algorithmica, 81(10):3890-3935, 2019. URL:
  11. Édouard Bonnet, Michael Lampis, and Vangelis Th. Paschos. Time-approximation trade-offs for inapproximable problems. J. Comput. Syst. Sci., 92:171-180, 2018. URL:
  12. Nicolas Boria, Federico Della Croce, and Vangelis Th. Paschos. On the max min vertex cover problem. Discret. Appl. Math., 196:62-71, 2015. URL:
  13. Katrin Casel, Henning Fernau, Mehdi Khosravian Ghadikolaei, Jérôme Monnot, and Florian Sikora. On the complexity of solution extension of optimization problems. Theor. Comput. Sci., 904:48-65, 2022. URL:
  14. Juhi Chaudhary, Sounaka Mishra, and B. S. Panda. Minimum maximal acyclic matching in proper interval graphs. In CALDAM, volume 13947 of Lecture Notes in Computer Science, pages 377-388. Springer, 2023. URL:
  15. Maria Chudnovsky and Paul D. Seymour. The three-in-a-tree problem. Comb., 30(4):387-417, 2010. URL:
  16. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  17. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. ACM Trans. Algorithms, 18(2):17:1-17:31, 2022. URL:
  18. Reinhard Diestel. Graph Theory, volume 173 of Graduate texts in mathematics. Springer, 2017. URL:
  19. Gabriel L. Duarte, Hiroshi Eto, Tesshu Hanaka, Yasuaki Kobayashi, Yusuke Kobayashi, Daniel Lokshtanov, Lehilton L. C. Pedrosa, Rafael C. S. Schouery, and Uéverton S. Souza. Computing the largest bond and the maximum connected cut of a graph. Algorithmica, 83(5):1421-1458, 2021. URL:
  20. Louis Dublois, Tesshu Hanaka, Mehdi Khosravian Ghadikolaei, Michael Lampis, and Nikolaos Melissinos. (in)approximability of maximum minimal FVS. J. Comput. Syst. Sci., 124:26-40, 2022. URL:
  21. Louis Dublois, Michael Lampis, and Vangelis Th. Paschos. Upper dominating set: Tight algorithms for pathwidth and sub-exponential approximation. Theor. Comput. Sci., 923:271-291, 2022. URL:
  22. Fabio Furini, Ivana Ljubic, and Markus Sinnl. An effective dynamic programming algorithm for the minimum-cost maximal knapsack packing problem. Eur. J. Oper. Res., 262(2):438-448, 2017. URL:
  23. Ajinkya Gaikwad, Hitendra Kumar, Soumen Maity, Saket Saurabh, and Shuvam Kant Tripathi. Maximum minimal feedback vertex set: A parameterized perspective. CoRR, abs/2208.01953, 2022. URL:
  24. Laurent Gourvès, Jérôme Monnot, and Aris Pagourtzis. The lazy bureaucrat problem with common arrivals and deadlines: Approximation and mechanism design. In FCT, volume 8070 of Lecture Notes in Computer Science, pages 171-182. Springer, 2013. URL:
  25. Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics: A Foundation for Computer Science, 2nd Ed. Addison-Wesley, 1994. Google Scholar
  26. Tesshu Hanaka, Yasuaki Kobayashi, Yusuke Kobayashi, and Tsuyoshi Yagita. Finding a maximum minimal separator: Graph classes and fixed-parameter tractability. Theor. Comput. Sci., 865:131-140, 2021. URL:
  27. Ararat Harutyunyan, Michael Lampis, and Nikolaos Melissinos. Digraph coloring and distance to acyclicity. In STACS, volume 187 of LIPIcs, pages 41:1-41:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL:
  28. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL:
  29. Kai-Yuan Lai, Hsueh-I Lu, and Mikkel Thorup. Three-in-a-tree in near linear time. In STOC, pages 1279-1292. ACM, 2020. URL:
  30. Michael Lampis. Minimum stable cut and treewidth. In ICALP, volume 198 of LIPIcs, pages 92:1-92:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL:
  31. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. SIAM J. Comput., 47(3):675-702, 2018. URL:
  32. Sounaka Mishra and Kripasindhu Sikdar. On the hardness of approximating some np-optimization problems related to minimum linear ordering problem. RAIRO Theor. Informatics Appl., 35(3):287-309, 2001. URL:
  33. Michal Pilipczuk. Problems parameterized by treewidth tractable in single exponential time: A logical approach. In MFCS, volume 6907 of Lecture Notes in Computer Science, pages 520-531. Springer, 2011. URL:
  34. Meirav Zehavi. Maximum minimal vertex cover parameterized by vertex cover. SIAM J. Discret. Math., 31(4):2440-2456, 2017. URL:
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