Skip to main content
SpringerLink
Log in

An Improved Deterministic Parameterized Algorithm for Cactus Vertex Deletion

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

A cactus is a connected graph that does not contain K4 − e as a minor. Given a graph G = (V,E) and an integer k ≥ 0, Cactus Vertex Deletion (also known as Diamond Hitting Set) is the problem of deciding whether G has a vertex set of size at most k whose removal leaves a forest of cacti. The previously best deterministic parameterized algorithm for this problem was due to Bonnet et al. [WG 2016], which runs in time 26knO(1), where n is the number of vertices of G. In this paper, we design a deterministic algorithm for Cactus Vertex Deletion, which runs in time 17.64knO(1). As an almost straightforward application of our algorithm, we also give a deterministic 17.64knO(1)-time algorithm for Even Cycle Transversal, which improves the previous running time 50knO(1) of the known deterministic parameterized algorithm due to Misra et al. [WG 2012].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. The notation O∗ suppresses a polynomial factor of the input size.

  2. Note that Branching rules 3 and 4 are still valid for any triangle B in G[V ∖ S]. However, for ease of the case analysis, we focus on a leaf block of size 3 here.

References

  1. Becker, A., Bar-Yehuda, R., Geiger, D.: Randomized algorithms for the loop cutset problem. J. Artif. Int. Res. 12(1), 219–234 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Bodlaender, H.L., Ono, H., Otachi, Y.: A faster parameterized algorithm for Pseudoforest Deletion. Discret. Appl. Math. 236, 42–56 (2018). https://doi.org/10.1016/j.dam.2017.10.018

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonnet, É., Brettell, N., Kwon, O., Marx, D.: Parameterized vertex deletion problems for hereditary graph classes with a block property. In: Heggernes, P. (ed.) Proceedings of WG 2016, Lecture Notes in Computer Science, vol. 9941, pp 233–244 (2016), https://doi.org/10.1007/978-3-662-53536-3_20

  4. Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40-42), 3736–3756 (2010). https://doi.org/10.1016/j.tcs.2010.06.026

    Article  MathSciNet  MATH  Google Scholar 

  5. Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., Rooij, J.M.M.V., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS ’11, pp 150–159. IEEE Computer Society, USA (2011), https://doi.org/10.1109/FOCS.2011.23

  6. Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM 61 (4), 23:1–23:27 (2014)

    Article  MathSciNet  Google Scholar 

  7. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness i: basic results. SIAM J. Comput. 24(4), 873–921 (1995). https://doi.org/10.1137/S0097539792228228

    Article  MathSciNet  MATH  Google Scholar 

  8. Feng, Q., Wang, J., Li, S., Chen, J.: Randomized parameterized algorithms for p2-packing and co-path packing problems. J. Comb. Optim. 29(1), 125–140 (2015). https://doi.org/10.1007/s10878-013-9691-z

    Article  MathSciNet  MATH  Google Scholar 

  9. Fiorini, S., Joret, G., Pietropaoli, U.: Hitting diamonds and growing Cacti. In: Eisenbrand, F., Shepherd, F.B. (eds.) Proceedings of IPCO 2010, Lecture Notes in Computer Science, vol. 6080, pp 191–204. Springer (2010), https://doi.org/10.1007/978-3-642-13036-6_15

  10. Fomin, F.V., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. J. ACM 56(5), 25:1–25:32 (2009). https://doi.org/10.1145/1552285.1552286

    Article  MathSciNet  MATH  Google Scholar 

  11. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms, 1st edn. Springer, Berlin (2010)

    Book  Google Scholar 

  12. Fomin, F.V., Lokshtanov, D., Misra, N., Philip, G., Saurabh, S.: Hitting forbidden minors: Approximation and kernelization. SIAM J. Discret. Math. 30(1), 383–410 (2016). https://doi.org/10.1137/140997889

    Article  MathSciNet  MATH  Google Scholar 

  13. Gowda, K.N., Lonkar, A., Panolan, F., Patel, V., Saurabh, S.: Improved FPT algorithms for deletion to forest-like structures. In: Cao, Y., Cheng, S., Li, M. (eds.) 31st International Symposium on Algorithms and Computation, ISAAC 2020, December 14-18, 2020, Hong Kong, China (Virtual Conference), LIPIcs, vol. 181, pp 34:1–34:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020), https://doi.org/10.4230/LIPIcs.ISAAC.2020.34

  14. Guo, J., Gramm, J., Hüffner, F., Niedermeier, R., Wernicke, S.: Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J. Comput. Syst. Sci. 72(8), 1386–1396 (2006). https://doi.org/10.1016/j.jcss.2006.02.001

    Article  MathSciNet  MATH  Google Scholar 

  15. Iwata, Y., Kobayashi, Y.: Improved analysis of Highest-Degree branching for feedback vertex set. In: Jansen, B.M.P., Telle, J.A. (eds.) Proceedings of IPEC 2019, LIPIcs, vol. 148, pp 22:1–22:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019), https://doi.org/10.4230/LIPIcs.IPEC.2019.22

  16. Kolay, S., Lokshtanov, D., Panolan, F., Saurabh, S.: Quick but odd growth of cacti. Algorithmica 79(1), 271–290 (2017). https://doi.org/10.1007/s00453-017-0317-1

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, J., Nederlof, J.: Detecting feedback vertex sets of size k in o ∗ (2.7k time. In: Chawla, S. (ed.) Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pp 971–989. SIAM (2020), https://doi.org/10.1137/1.9781611975994.58

  18. Misra, P., Raman, V., Ramanujan, M.S., Saurabh, S.: Parameterized algorithms for even cycle transversal. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds.) Proceedings of WG 2012, Lecture Notes in Computer Science, vol. 7551, pp 172–183. Springer, Berlin (2012), https://doi.org/10.1007/978-3-642-34611-8_19

  19. Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004). https://doi.org/10.1016/j.orl.2003.10.009

    Article  MathSciNet  MATH  Google Scholar 

  20. Xiao, M.: A parameterized algorithm for bounded-degree vertex deletion. In: Dinh, T.N., Thai, M.T. (eds.) Proceedings of COCOON 2016, Lecture Notes in Computer Science, vol. 9797, pp 79–91. Springer (2016), https://doi.org/10.1007/978-3-319-42634-1_7

Download references

Author information

Authors and Affiliations

  1. School of Data Science, Yokohama City University, Yokohama, Japan

    Yuuki Aoike

  2. Graduate School of Informatics, Nagoya University, Nagoya, Japan

    Tatsuya Gima, Tesshu Hanaka & Yota Otachi

  3. Faculty of Science and Technology, Seikei University, Seikei, Japan

    Masashi Kiyomi

  4. Graduate School of Informatics, Kyoto University, Kyoto, Japan

    Yasuaki Kobayashi

  5. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

    Yusuke Kobayashi

  6. National Institute of Informatics, Chiyoda-ku, Tokyo, Japan

    Kazuhiro Kurita

Authors
  1. Yuuki Aoike
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatsuya Gima
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tesshu Hanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Masashi Kiyomi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yasuaki Kobayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Yusuke Kobayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Kazuhiro Kurita
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Yota Otachi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yasuaki Kobayashi.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP18H05291, JP18K11168, JP18K11169, JP19K21537, JP20H05793, JP20K11692, and JP20K19742. The authors thank Kunihiro Wasa for fruitful discussions.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aoike, Y., Gima, T., Hanaka, T. et al. An Improved Deterministic Parameterized Algorithm for Cactus Vertex Deletion. Theory Comput Syst 66, 502–515 (2022). https://doi.org/10.1007/s00224-022-10076-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-022-10076-x

Keywords

Search

Navigation