Document

# Close Relatives (Of Feedback Vertex Set), Revisited

## File

LIPIcs.IPEC.2021.21.pdf
• Filesize: 1 MB
• 15 pages

## Cite As

Hugo Jacob, Thomas Bellitto, Oscar Defrain, and Marcin Pilipczuk. Close Relatives (Of Feedback Vertex Set), Revisited. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.IPEC.2021.21

## Abstract

At IPEC 2020, Bergougnoux, Bonnet, Brettell, and Kwon (Close Relatives of Feedback Vertex Set Without Single-Exponential Algorithms Parameterized by Treewidth, IPEC 2020, LIPIcs vol. 180, pp. 3:1-3:17) showed that a number of problems related to the classic Feedback Vertex Set (FVS) problem do not admit a 2^{o(k log k)} ⋅ n^{𝒪(1)}-time algorithm on graphs of treewidth at most k, assuming the Exponential Time Hypothesis. This contrasts with the 3^{k} ⋅ k^{𝒪(1)} ⋅ n-time algorithm for FVS using the Cut&Count technique. During their live talk at IPEC 2020, Bergougnoux et al. posed a number of open questions, which we answer in this work. - Subset Even Cycle Transversal, Subset Odd Cycle Transversal, Subset Feedback Vertex Set can be solved in time 2^{𝒪(k log k)} ⋅ n in graphs of treewidth at most k. This matches a lower bound for Even Cycle Transversal of Bergougnoux et al. and improves the polynomial factor in some of their upper bounds. - Subset Feedback Vertex Set and Node Multiway Cut can be solved in time 2^{𝒪(k log k)} ⋅ n, if the input graph is given as a cliquewidth expression of size n and width k. - Odd Cycle Transversal can be solved in time 4^k ⋅ k^{𝒪(1)} ⋅ n if the input graph is given as a cliquewidth expression of size n and width k. Furthermore, the existence of a constant ε > 0 and an algorithm performing this task in time (4-ε)^k ⋅ n^{𝒪(1)} would contradict the Strong Exponential Time Hypothesis. A common theme of the first two algorithmic results is to represent connectivity properties of the current graph in a state of a dynamic programming algorithm as an auxiliary forest with 𝒪(k) nodes. This results in a 2^{𝒪(k log k)} bound on the number of states for one node of the tree decomposition or cliquewidth expression and allows to compare two states in k^{𝒪(1)} time, resulting in linear time dependency on the size of the graph or the input cliquewidth expression.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
##### Keywords
• feedback vertex set
• treewidth
• cliquewidth

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. Hitting minors on bounded treewidth graphs. IV. an optimal algorithm. CoRR, abs/1907.04442, 2019. URL: http://arxiv.org/abs/1907.04442.
2. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. Hitting minors on bounded treewidth graphs. I. general upper bounds. SIAM J. Discret. Math., 34(3):1623-1648, 2020. URL: https://doi.org/10.1137/19M1287146.
3. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. Hitting minors on bounded treewidth graphs. II. single-exponential algorithms. Theor. Comput. Sci., 814:135-152, 2020. URL: https://doi.org/10.1016/j.tcs.2020.01.026.
4. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. Hitting minors on bounded treewidth graphs. III. lower bounds. J. Comput. Syst. Sci., 109:56-77, 2020. URL: https://doi.org/10.1016/j.jcss.2019.11.002.
5. Benjamin Bergougnoux, Édouard Bonnet, Nick Brettell, and O-joung Kwon. Close Relatives of Feedback Vertex Set Without Single-Exponential Algorithms Parameterized by Treewidth. In Yixin Cao and Marcin Pilipczuk, editors, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020), volume 180 of Leibniz International Proceedings in Informatics (LIPIcs), pages 3:1-3:17, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.IPEC.2020.3.
6. Benjamin Bergougnoux and Mamadou Moustapha Kanté. Fast exact algorithms for some connectivity problems parameterized by clique-width. Theor. Comput. Sci., 782:30-53, 2019. URL: https://doi.org/10.1016/j.tcs.2019.02.030.
7. 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: https://doi.org/10.1016/j.ic.2014.12.008.
8. Marthe Bonamy, Lukasz Kowalik, Jesper Nederlof, Michal Pilipczuk, Arkadiusz Socala, and Marcin Wrochna. On directed feedback vertex set parameterized by treewidth. In Andreas Brandstädt, Ekkehard Köhler, and Klaus Meer, editors, Graph-Theoretic Concepts in Computer Science - 44th International Workshop, WG 2018, Cottbus, Germany, June 27-29, 2018, Proceedings, volume 11159 of Lecture Notes in Computer Science, pages 65-78. Springer, 2018. URL: https://doi.org/10.1007/978-3-030-00256-5_6.
9. Binh-Minh Bui-Xuan, Ondrej Suchý, Jan Arne Telle, and Martin Vatshelle. Feedback vertex set on graphs of low clique-width. Eur. J. Comb., 34(3):666-679, 2013. URL: https://doi.org/10.1016/j.ejc.2012.07.023.
10. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
11. Bruno Courcelle, Joost Engelfriet, and Grzegorz Rozenberg. Handle-rewriting hypergraph grammars. J. Comput. Syst. Sci., 46(2):218-270, 1993. URL: https://doi.org/10.1016/0022-0000(93)90004-G.
12. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
13. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms, volume 5. Springer, 2015.
14. Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast hamiltonicity checking via bases of perfect matchings. J. ACM, 65(3):12:1-12:46, 2018. URL: https://doi.org/10.1145/3148227.
15. Marek Cygan, Dániel Marx, Marcin Pilipczuk, and Michal Pilipczuk. Hitting forbidden subgraphs in graphs of bounded treewidth. Inf. Comput., 256:62-82, 2017. URL: https://doi.org/10.1016/j.ic.2017.04.009.
16. 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. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 150-159. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.23.
17. Samuel Fiorini, Nadia Hardy, Bruce A. Reed, and Adrian Vetta. Planar graph bipartization in linear time. Discret. Appl. Math., 156(7):1175-1180, 2008. URL: https://doi.org/10.1016/j.dam.2007.08.013.
18. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Intractability of clique-width parameterizations. SIAM J. Comput., 39(5):1941-1956, 2010. URL: https://doi.org/10.1137/080742270.
19. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Almost optimal lower bounds for problems parameterized by clique-width. SIAM J. Comput., 43(5):1541-1563, 2014. URL: https://doi.org/10.1137/130910932.
20. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Clique-width III: hamiltonian cycle and the odd case of graph coloring. ACM Trans. Algorithms, 15(1):9:1-9:27, 2019. URL: https://doi.org/10.1145/3280824.
21. Ton Kloks. Treewidth: computations and approximations, volume 842. Springer Science & Business Media, 1994.
22. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 777-789. SIAM, 2011.
23. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. In Dana Randall, editor, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 760-776. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.60.
24. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. ACM Trans. Algorithms, 14(2):13:1-13:30, 2018. URL: https://doi.org/10.1145/3170442.
25. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. SIAM J. Comput., 47(3):675-702, 2018. URL: https://doi.org/10.1137/16M1104834.
26. Pranabendu Misra, Venkatesh Raman, MS Ramanujan, and Saket Saurabh. Parameterized algorithms for even cycle transversal. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 172-183. Springer, 2012.
27. Michal Pilipczuk. Problems parameterized by treewidth tractable in single exponential time: A logical approach. In Filip Murlak and Piotr Sankowski, editors, Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Warsaw, Poland, August 22-26, 2011. Proceedings, volume 6907 of Lecture Notes in Computer Science, pages 520-531. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22993-0_47.
28. Neil Robertson and Paul D. Seymour. Graph minors. III. planar tree-width. J. Comb. Theory, Ser. B, 36(1):49-64, 1984. URL: https://doi.org/10.1016/0095-8956(84)90013-3.
29. Ignasi Sau and Uéverton dos Santos Souza. Hitting forbidden induced subgraphs on bounded treewidth graphs. In Javier Esparza and Daniel Král', editors, 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, August 24-28, 2020, Prague, Czech Republic, volume 170 of LIPIcs, pages 82:1-82:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.82.
30. Egon Wanke. k-nlc graphs and polynomial algorithms. Discret. Appl. Math., 54(2-3):251-266, 1994. URL: https://doi.org/10.1016/0166-218X(94)90026-4.
X

Feedback for Dagstuhl Publishing