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On Kernels for d-Path Vertex Cover

Authors Radovan Červený , Pratibha Choudhary , Ondřej Suchý



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Author Details

Radovan Červený
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
Pratibha Choudhary
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
Ondřej Suchý
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic

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Radovan Červený, Pratibha Choudhary, and Ondřej Suchý. On Kernels for d-Path Vertex Cover. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 29:1-29:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.29

Abstract

In this paper we study the kernelization of the d-Path Vertex Cover (d-PVC) problem. Given a graph G, the problem requires finding whether there exists a set of at most k vertices whose removal from G results in a graph that does not contain a path (not necessarily induced) with d vertices. It is known that d-PVC is NP-complete for d ≥ 2. Since the problem generalizes to d-Hitting Set, it is known to admit a kernel with 𝒪(dk^d) edges. We improve on this by giving better kernels. Specifically, we give kernels with 𝒪(k²) vertices and edges for the cases when d = 4 and d = 5. Further, we give a kernel with 𝒪(k⁴d^{2d+9}) vertices and edges for general d.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Fixed parameter tractability
Keywords
  • Parameterized complexity
  • Kernelization
  • d-Hitting Set
  • d-Path Vertex Cover
  • Expansion Lemma

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References

  1. Boštjan Brešar, František Kardoš, Ján Katrenič, and Gabriel Semanišin. Minimum k-path vertex cover. Discrete Applied Mathematics, 159(12):1189-1195, 2011. URL: https://doi.org/10.1016/j.dam.2011.04.008.
  2. Eglantine Camby, Jean Cardinal, Mathieu Chapelle, Samuel Fiorini, and Gwenaël Joret. A primal-dual 3-approximation algorithm for hitting 4-vertex paths. In 9th International colloquium on graph theory and combinatorics, 2014. Google Scholar
  3. Radovan Červený and Ondřej Suchý. Faster FPT algorithm for 5-path vertex cover. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, volume 138 of LIPIcs, pages 32:1-32:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.MFCS.2019.32.
  4. Radovan Červený and Ondřej Suchý. Faster FPT algorithm for 5-path vertex cover. CoRR, abs/1906.09213, 2019. URL: http://arxiv.org/abs/1906.09213.
  5. Radovan Červený and Ondřej Suchý. Generating faster algorithms for d-path vertex cover. CoRR, abs/2111.05896, 2021. URL: http://arxiv.org/abs/2111.05896.
  6. Maw-Shang Chang, Li-Hsuan Chen, Ling-Ju Hung, Yi-Zhi Liu, Peter Rossmanith, and Somnath Sikdar. Moderately exponential time algorithms for the maximum bounded-degree-1 set problem. Discret. Appl. Math., 251:114-125, 2018. URL: https://doi.org/10.1016/j.dam.2018.05.032.
  7. 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.
  8. 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.
  9. Holger Dell and Dániel Marx. Kernelization of packing problems. CoRR, abs/1812.03155, 2018. URL: http://arxiv.org/abs/1812.03155.
  10. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 251-260. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806725.
  11. Reinhard Diestel. Graph Theory, 5th Edition, volume 173 of Graduate texts in mathematics. Springer, 2016. Google Scholar
  12. Jack Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17:449-467, 1965. URL: https://doi.org/10.4153/CJM-1965-045-4.
  13. Stefan Fafianie and Stefan Kratsch. A shortcut to (sun)flowers: Kernels in logarithmic space or linear time. In Giuseppe F. Italiano, Giovanni Pighizzini, and Donald Sannella, editors, Mathematical Foundations of Computer Science 2015 - 40th International Symposium, MFCS 2015, Milan, Italy, August 24-28, 2015, Proceedings, Part II, volume 9235 of Lecture Notes in Computer Science, pages 299-310. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48054-0_25.
  14. Henning Fernau. Parameterized algorithmics for d-hitting set. Int. J. Comput. Math., 87(14):3157-3174, 2010. URL: https://doi.org/10.1080/00207160903176868.
  15. Fedor V. Fomin, Serge Gaspers, Dieter Kratsch, Mathieu Liedloff, and Saket Saurabh. Iterative compression and exact algorithms. Theor. Comput. Sci., 411(7-9):1045-1053, 2010. URL: https://doi.org/10.1016/j.tcs.2009.11.012.
  16. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and kernelization. SIAM J. Discret. Math., 30(1):383-410, 2016. URL: https://doi.org/10.1137/140997889.
  17. 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), September 2016. URL: https://doi.org/10.1145/2886094.
  18. Stefan Funke, André Nusser, and Sabine Storandt. On k-path covers and their applications. VLDB J., 25(1):103-123, 2016. URL: https://doi.org/10.1007/s00778-015-0392-3.
  19. Michael Lampis. A kernel of order 2k-clog k for vertex cover. Inf. Process. Lett., 111(23-24):1089-1091, 2011. URL: https://doi.org/10.1016/j.ipl.2011.09.003.
  20. Euiwoong Lee. Partitioning a graph into small pieces with applications to path transversal. Math. Program., 177(1-2):1-19, 2019. URL: https://doi.org/10.1007/s10107-018-1255-7.
  21. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci., 20(2):219-230, 1980. URL: https://doi.org/10.1016/0022-0000(80)90060-4.
  22. Marián Novotný. Design and analysis of a generalized canvas protocol. In Proc. 4th IFIP WG 11.2 International Workshop on Information Security Theory and Practices. Security and Privacy of Pervasive Systems and Smart Devices, WISTP 2010, pages 106-121, 2010. URL: https://doi.org/10.1007/978-3-642-12368-9_8.
  23. Elena Prieto-Rodríguez. Systematic kernelization in FPT algorithm design. PhD thesis, University of Newcastle, 2005. URL: http://hdl.handle.net/1959.13/1418337.
  24. Stéphan Thomassé. A 4k² kernel for feedback vertex set. ACM Trans. Algorithms, 6(2):32:1-32:8, 2010. URL: https://doi.org/10.1145/1721837.1721848.
  25. Dekel Tsur. Faster deterministic parameterized algorithm for k-path. Theor. Comput. Sci., 790:96-104, 2019. URL: https://doi.org/10.1016/j.tcs.2019.04.024.
  26. Dekel Tsur. l-path vertex cover is easier than l-hitting set for small l. CoRR, abs/1906.10523, 2019. URL: http://arxiv.org/abs/1906.10523.
  27. Dekel Tsur. Parameterized algorithm for 3-path vertex cover. Theor. Comput. Sci., 783:1-8, 2019. URL: https://doi.org/10.1016/j.tcs.2019.03.013.
  28. Dekel Tsur. An O^*(2.619^k) algorithm for 4-path vertex cover. Discret. Appl. Math., 291:1-14, 2021. URL: https://doi.org/10.1016/j.dam.2020.11.019.
  29. Jianhua Tu. A survey on the k-path vertex cover problem. CoRR, abs/2201.03397, 2022. URL: http://arxiv.org/abs/2201.03397.
  30. Jianhua Tu and Wenli Zhou. A primal-dual approximation algorithm for the vertex cover P₃ problem. Theor. Comput. Sci., 412(50):7044-7048, 2011. URL: https://doi.org/10.1016/j.tcs.2011.09.013.
  31. Mingyu Xiao and Shaowei Kou. Exact algorithms for the maximum dissociation set and minimum 3-path vertex cover problems. Theor. Comput. Sci., 657:86-97, 2017. URL: https://doi.org/10.1016/j.tcs.2016.04.043.
  32. Mingyu Xiao and Shaowei Kou. Kernelization and parameterized algorithms for 3-path vertex cover. In T. V. Gopal, Gerhard Jäger, and Silvia Steila, editors, Theory and Applications of Models of Computation - 14th Annual Conference, TAMC 2017, Bern, Switzerland, April 20-22, 2017, Proceedings, volume 10185 of Lecture Notes in Computer Science, pages 654-668, 2017. URL: https://doi.org/10.1007/978-3-319-55911-7_47.
  33. Mingyu Xiao and Hiroshi Nagamochi. Exact algorithms for maximum independent set. Inf. Comput., 255:126-146, 2017. URL: https://doi.org/10.1016/j.ic.2017.06.001.
  34. Meirav Zehavi. Mixing color coding-related techniques. In Nikhil Bansal and Irene Finocchi, editors, Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, volume 9294 of Lecture Notes in Computer Science, pages 1037-1049. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_86.
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