Document Open Access Logo

Component Order Connectivity Admits No Polynomial Kernel Parameterized by the Distance to Subdivided Comb Graphs

Authors Jakob Greilhuber , Roohani Sharma

PDF
Thumbnail PDF

File

LIPIcs.IPEC.2024.21.pdf
  • Filesize: 0.92 MB
  • 17 pages

Document Identifiers

Author Details

Jakob Greilhuber
  • TU Wien, Austria
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
  • Saarbrücken Graduate School of Computer Science, Germany
Roohani Sharma
  • Department of Informatics, University of Bergen, Norway

Acknowledgements

The majority of the work was done at the Max Planck Institute of Informatics during a summer internship of Jakob Greilhuber. We thank the anonymous reviewers that reviewed a previous version of this paper. The presentation of this work is based on their comments. Jakob Greilhuber thanks the organizers of ALGO 2024 for their financial support.

Cite As Get BibTex

Jakob Greilhuber and Roohani Sharma. Component Order Connectivity Admits No Polynomial Kernel Parameterized by the Distance to Subdivided Comb Graphs. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.21

Abstract

In this paper we show that the d-Component Order Connectivity (d-COC) problem parameterized by the distance to subdivided comb graphs (dist-to-subdivided-combs) does not admit a polynomial kernel, unless NP ⊆ coNP/poly. 
The d-COC problem is a generalization of the classical Vertex Cover problem. An instance of the d-COC problem consists of an undirected graph G and a positive integer k, and the question is whether there exists a set S ⊆ V(G) of size at most k, such that each connected component of G-S contains at most d vertices. When d = 1, d-COC is the Vertex Cover problem.
Vertex Cover is a ubiquitous problem in parameterized complexity, and it admits a kernel with O(k²) edges and O(k) vertices, which is tight [Dell & van Melkebeek, JACM 2014]. Our result is inspired by the work of Jansen & Bodlaender [TOCS 2013], who gave the first polynomial kernel for Vertex Cover where the parameter is provably smaller or equal to the standard parameter, solution size k. They used fvs, the feedback vertex set number, as the parameter. In this work, we show that unlike most other existing results or techniques for kernelization of Vertex Cover that generalize to d-COC, this is not the case when dist-to-subdivided-combs, which is at least as large as fvs, is the parameter.
Our lower bound is achieved in two stages. In the first stage we extend the result of Hols, Kratsch & Pieterse [SIDMA 2022] where they show that if a graph family 𝒞, which is closed under taking disjoint unions, has unbounded "blocking set" size, then Vertex Cover does not admit a polynomial kernel parameterized by the size of a vertex modulator to 𝒞, unless NP ⊆ coNP/poly. We show that a similar sufficient condition for proving the non-existence of polynomial kernels also holds for d-COC. In the second stage, we show that when 𝒞 is the family of subdivided comb graphs, contrary to Vertex Cover, where the size of minimal blocking sets of graphs in 𝒞 is at most two [Jansen & Bodlaender, STACS 2011], the size of minimal blocking sets of graphs in 𝒞 for the d-COC problem can be arbitrarily large. This yields the desired lower bound. In addition to this we also show that when 𝒞 is a class of paths, then it still has blocking sets of size at most two for d-COC, indicating that polynomial kernels might be achievable when the parameter is the size of a vertex modulator to the class of disjoint unions of paths (linear forests).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Component Order Connectivity
  • Kernelization
  • Structural Parameterizations
  • Feedback Vertex Set
  • Vertex Cover
  • Blocking Sets
  • Subdivided Comb Graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Faisal N. Abu-Khzam, Rebecca L. Collins, Michael R. Fellows, Michael A. Langston, W. Henry Suters, and Christopher T. Symons. Kernelization algorithms for the vertex cover problem: Theory and experiments. In Lars Arge, Giuseppe F. Italiano, and Robert Sedgewick, editors, Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algorithmics and Combinatorics, New Orleans, LA, USA, January 10, 2004, pages 62-69. SIAM, 2004. Google Scholar
  2. R. Balasubramanian, Michael R. Fellows, and Venkatesh Raman. An improved fixed-parameter algorithm for vertex cover. Inf. Process. Lett., 65(3):163-168, 1998. URL: https://doi.org/10.1016/S0020-0190(97)00213-5.
  3. Sriram Bhyravarapu, Satyabrata Jana, Saket Saurabh, and Roohani Sharma. Difference determines the degree: Structural kernelizations of component order connectivity. In Neeldhara Misra and Magnus Wahlström, editors, 18th International Symposium on Parameterized and Exact Computation, IPEC 2023, September 6-8, 2023, Amsterdam, The Netherlands, volume 285 of LIPIcs, pages 5:1-5:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.IPEC.2023.5.
  4. Marin Bougeret, Bart M. P. Jansen, and Ignasi Sau. Bridge-depth characterizes which minor-closed structural parameterizations of vertex cover admit a polynomial kernel. SIAM J. Discret. Math., 36(4):2737-2773, 2022. URL: https://doi.org/10.1137/21M1400766.
  5. Marin Bougeret and Ignasi Sau. How much does a treedepth modulator help to obtain polynomial kernels beyond sparse graphs? Algorithmica, 81(10):4043-4068, 2019. URL: https://doi.org/10.1007/S00453-018-0468-8.
  6. Jonathan F. Buss and Judy Goldsmith. Nondeterminism within P. SIAM J. Comput., 22(3):560-572, 1993. URL: https://doi.org/10.1137/0222038.
  7. Jianer Chen, Iyad A. Kanj, and Weijia Jia. Vertex cover: Further observations and further improvements. J. Algorithms, 41(2):280-301, 2001. URL: https://doi.org/10.1006/JAGM.2001.1186.
  8. Jianer Chen, Iyad A. Kanj, and Ge Xia. Improved upper bounds for vertex cover. Theor. Comput. Sci., 411(40-42):3736-3756, 2010. URL: https://doi.org/10.1016/J.TCS.2010.06.026.
  9. Jianer Chen, Lihua Liu, and Weijia Jia. Improvement on vertex cover for low-degree graphs. Networks, 35(4):253-259, 2000. URL: https://doi.org/10.1002/1097-0037(200007)35:4<253::AID-NET3>3.0.CO;2-K.
  10. Benny Chor, Mike Fellows, and David W. Juedes. Linear kernels in linear time, or how to save k colors in O(n²) steps. In Juraj Hromkovic, Manfred Nagl, and Bernhard Westfechtel, editors, Graph-Theoretic Concepts in Computer Science, 30th International Workshop,WG 2004, Bad Honnef, Germany, June 21-23, 2004, Revised Papers, volume 3353 of Lecture Notes in Computer Science, pages 257-269. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-30559-0_22.
  11. 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.
  12. 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.
  13. Frank K. H. A. Dehne, Michael R. Fellows, Frances A. Rosamond, and Peter Shaw. Greedy localization, iterative compression, modeled crown reductions: New FPT techniques, an improved algorithm for set splitting, and a novel 2k kernelization for vertex cover. In Rodney G. Downey, Michael R. Fellows, and Frank K. H. A. Dehne, editors, Parameterized and Exact Computation, First International Workshop, IWPEC 2004, Bergen, Norway, September 14-17, 2004, Proceedings, volume 3162 of Lecture Notes in Computer Science, pages 271-280. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-28639-4_24.
  14. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. Journal of The Acm, 61(4):23:1-23:27, 2014. URL: https://doi.org/10.1145/2629620.
  15. Huib Donkers and Bart M. P. Jansen. A turing kernelization dichotomy for structural parameterizations of F-minor-free deletion. J. Comput. Syst. Sci., 119:164-182, 2021. URL: https://doi.org/10.1016/J.JCSS.2021.02.005.
  16. Rodney G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness I: basic results. SIAM J. Comput., 24(4):873-921, 1995. URL: https://doi.org/10.1137/S0097539792228228.
  17. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  18. Pål Grønås Drange, Markus S. Dregi, and Pim van 't Hof. On the computational complexity of vertex integrity and component order connectivity. Algorithmica, 76(4):1181-1202, 2016. URL: https://doi.org/10.1007/S00453-016-0127-X.
  19. Michael R. Fellows. Blow-ups, win/win’s, and crown rules: Some new directions in FPT. In Hans L. Bodlaender, editor, Graph-Theoretic Concepts in Computer Science, 29th International Workshop, WG 2003, Elspeet, The Netherlands, June 19-21, 2003, Revised Papers, volume 2880 of Lecture Notes in Computer Science, pages 1-12. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-39890-5_1.
  20. Michael R. Fellows, Lars Jaffke, Aliz Izabella Király, Frances A. Rosamond, and Mathias Weller. What is known about vertex cover kernelization? In Hans-Joachim Böckenhauer, Dennis Komm, and Walter Unger, editors, Adventures between Lower Bounds and Higher Altitudes - Essays Dedicated to Juraj Hromkovič on the Occasion of His 60th Birthday, volume 11011 of Lecture Notes in Computer Science, pages 330-356. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-98355-4_19.
  21. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of parameterized preprocessing. Cambridge University Press, 2019. URL: https://doi.org/10.1017/9781107415157.
  22. Fedor V. Fomin and Torstein J. F. Strømme. Vertex cover structural parameterization revisited. In Pinar Heggernes, editor, Graph-Theoretic Concepts in Computer Science - 42nd International Workshop, WG 2016, Istanbul, Turkey, June 22-24, 2016, Revised Selected Papers, volume 9941 of Lecture Notes in Computer Science, pages 171-182, 2016. URL: https://doi.org/10.1007/978-3-662-53536-3_15.
  23. Daniel Gross, Monika Heinig, Lakshmi Iswara, L William Kazmierczak, Kristi Luttrell, John T Saccoman, and Charles Suffel. A survey of component order connectivity models of graph theoretic networks. WSEAS Transactions on Mathematics, 12(9):895-910, 2013. Google Scholar
  24. David G. Harris and N. S. Narayanaswamy. A faster algorithm for vertex cover parameterized by solution size. In Olaf Beyersdorff, Mamadou Moustapha Kanté, Orna Kupferman, and Daniel Lokshtanov, editors, 41st International Symposium on Theoretical Aspects of Computer Science, STACS 2024, March 12-14, 2024, Clermont-Ferrand, France, volume 289 of LIPIcs, pages 40:1-40:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.STACS.2024.40.
  25. Danny Hermelin and Xi Wu. Weak compositions and their applications to polynomial lower bounds for kernelization. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 104-113. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.9.
  26. Eva-Maria C. Hols, Stefan Kratsch, and Astrid Pieterse. Elimination distances, blocking sets, and kernels for vertex cover. SIAM J. Discret. Math., 36(3):1955-1990, 2022. URL: https://doi.org/10.1137/20M1335285.
  27. Bart M. P. Jansen and Hans L. Bodlaender. Vertex cover kernelization revisited: Upper and lower bounds for a refined parameter. In Thomas Schwentick and Christoph Dürr, editors, 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011, March 10-12, 2011, Dortmund, Germany, volume 9 of LIPIcs, pages 177-188. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2011. URL: https://doi.org/10.4230/LIPICS.STACS.2011.177.
  28. Bart M. P. Jansen and Hans L. Bodlaender. Vertex cover kernelization revisited - upper and lower bounds for a refined parameter. Theory Comput. Syst., 53(2):263-299, 2013. URL: https://doi.org/10.1007/S00224-012-9393-4.
  29. Bart M. P. Jansen and Astrid Pieterse. Polynomial kernels for hitting forbidden minors under structural parameterizations. Theor. Comput. Sci., 841:124-166, 2020. URL: https://doi.org/10.1016/J.TCS.2020.07.009.
  30. Bart MP Jansen. The power of data reduction: Kernels for fundamental graph problems. PhD thesis, Utrecht University, 2013. Google Scholar
  31. Samir Khuller. Algorithms column: the vertex cover problem. SIGACT News, 33(2):31-33, 2002. URL: https://doi.org/10.1145/564585.564598.
  32. Mithilesh Kumar and Daniel Lokshtanov. A 2lk kernel for l-component order connectivity. In Jiong Guo and Danny Hermelin, editors, 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, August 24-26, 2016, Aarhus, Denmark, volume 63 of LIPIcs, pages 20:1-20:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.IPEC.2016.20.
  33. Michael Lampis. A kernel of order 2 k-c log k for vertex cover. Inf. Process. Lett., 111(23-24):1089-1091, 2011. URL: https://doi.org/10.1016/J.IPL.2011.09.003.
  34. Diptapriyo Majumdar, Venkatesh Raman, and Saket Saurabh. Polynomial kernels for vertex cover parameterized by small degree modulators. Theory of Computing Systems, 62(8):1910-1951, 2018. URL: https://doi.org/10.1007/S00224-018-9858-1.
  35. N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. LP can be a cure for parameterized problems. In Christoph Dürr and Thomas Wilke, editors, 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France, volume 14 of LIPIcs, pages 338-349. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. URL: https://doi.org/10.4230/LIPICS.STACS.2012.338.
  36. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006. URL: https://doi.org/10.1093/ACPROF:OSO/9780198566076.001.0001.
  37. Rolf Niedermeier and Peter Rossmanith. Upper bounds for vertex cover further improved. In Christoph Meinel and Sophie Tison, editors, STACS 99, 16th Annual Symposium on Theoretical Aspects of Computer Science, Trier, Germany, March 4-6, 1999, Proceedings, volume 1563 of Lecture Notes in Computer Science, pages 561-570. Springer, 1999. URL: https://doi.org/10.1007/3-540-49116-3_53.
  38. 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.
  39. Arezou Soleimanfallah and Anders Yeo. A kernel of order 2k-c for vertex cover. Discret. Math., 311(10-11):892-895, 2011. URL: https://doi.org/10.1016/J.DISC.2011.02.014.
  40. Ulrike Stege and Michael Ralph Fellows. An improved fixed parameter tractable algorithm for vertex cover. Technical report/Departement Informatik, ETH Zürich, 318, 1999. URL: https://doi.org/10.3929/ethz-a-006653305.
  41. Mingyu Xiao. Linear kernels for separating a graph into components of bounded size. Journal of Computer and System Sciences, 88:260-270, 2017. URL: https://doi.org/10.1016/J.JCSS.2017.04.004.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact