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Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees

Authors Leon Kellerhals , Tomohiro Koana , Pascal Kunz



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Author Details

Leon Kellerhals
  • Faculty IV, Institute of Software Engineering and Theoretical Computer Science, Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Tomohiro Koana
  • Faculty IV, Institute of Software Engineering and Theoretical Computer Science, Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Pascal Kunz
  • Faculty IV, Institute of Software Engineering and Theoretical Computer Science, Algorithmics and Computational Complexity, Technische Universität Berlin, Germany

Acknowledgements

This work was initiated at the research retreat of the Algorithmics and Computational Complexity group, TU Berlin, in 2021.

Cite AsGet BibTex

Leon Kellerhals, Tomohiro Koana, and Pascal Kunz. Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 19:1-19:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.IPEC.2022.19

Abstract

Vertex Cover parameterized by the solution size k is the quintessential fixed-parameter tractable problem. FPT algorithms are most interesting when the parameter is small. Several lower bounds on k are well-known, such as the maximum size of a matching. This has led to a line of research on parameterizations of Vertex Cover by the difference of the solution size k and a lower bound. The most prominent cases for such lower bounds for which the problem is FPT are the matching number or the optimal fractional LP solution. We investigate parameterizations by the difference between k and other graph parameters including the feedback vertex number, the degeneracy, cluster deletion number, and treewidth with the goal of finding the border of fixed-parameter tractability for said difference parameterizations. We also consider similar parameterizations of the Feedback Vertex Set problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • parameterized complexity
  • vertex cover
  • feedback vertex set
  • above guarantee parameterization

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References

  1. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM Journal on Discrete Mathematics, 28(1):277-305, 2014. URL: https://doi.org/10.1137/120880240.
  2. Marin Bougeret, Bart M. P. Jansen, and Ignasi Sau. Bridge-depth characterizes which structural parameterizations of vertex cover admit a polynomial kernel. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP), pages 16:1-16:19, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.16.
  3. Kathie Cameron and Tracy Walker. The graphs with maximum induced matching and maximum matching the same size. Discrete Mathematics, 299(1-3):49-55, 2005. URL: https://doi.org/10.1016/j.disc.2004.07.022.
  4. Jianer Chen, Iyad A. Kanj, and Ge Xia. Improved upper bounds for vertex cover. Theoretical Computer Science, 411(40-42):3736-3756, 2010. URL: https://doi.org/10.1016/j.tcs.2010.06.026.
  5. Václav Chvátal and Peter J. Slater. A note on well-covered graphs. In John Gimbel, John W. Kennedy, and Louis V. Quintas, editors, Quo Vadis, Graph Theory?, pages 179-181. Elsevier, 1993. URL: https://doi.org/10.1016/S0167-5060(08)70387-X.
  6. 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.
  7. Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. On the hardness of losing width. Theory of Computing Systems, 54(1):73-82, 2014. URL: https://doi.org/10.1007/s00224-013-9480-1.
  8. Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. On multiway cut parameterized above lower bounds. ACM Transactions on Computation Theory, 5(1):3:1-3:11, 2013. URL: https://doi.org/10.1145/2462896.2462899.
  9. Erik D. Demaine and MohammadTaghi HajiAghayi. Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica, 28(1):19-36, 2008. URL: https://doi.org/10.1007/s00493-008-2140-4.
  10. Erik D Demaine, MohammadTaghi Hajiaghayi, Naomi Nishimura, Prabhakar Ragde, and Dimitrios M Thilikos. Approximation algorithms for classes of graphs excluding single-crossing graphs as minors. Journal of Computer and System Sciences, 69(2):166-195, 2004. URL: https://doi.org/10.1016/j.jcss.2003.12.001.
  11. Marc Demange and Tínaz Ekim. Efficient recognition of equimatchable graphs. Information Processing Letters, 114(1-2):66-71, 2014. URL: https://doi.org/10.1016/j.ipl.2013.08.002.
  12. Zakir Deniz, Tínaz Ekim, Tatiana Romina Hartinger, Martin Milanic, and Mordechai Shalom. On two extensions of equimatchable graphs. Discrete Optimization, 26:112-130, 2017. URL: https://doi.org/10.1016/j.disopt.2017.08.002.
  13. Reinhard Diestel. Graph Theory. Springer, 5th edition, 2016. URL: https://doi.org/10.1007/978-3-662-53622-3.
  14. Rod G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoretical Computer Science, 141(1-2):109-131, 1995. URL: https://doi.org/10.1016/0304-3975(94)00097-3.
  15. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  16. Márcio Antônio Duarte, Felix Joos, Lucia Draque Penso, Dieter Rautenbach, and Uéverton S. Souza. Maximum induced matchings close to maximum matchings. Theoretical Computer Science, 588:131-137, 2015. URL: https://doi.org/10.1016/j.tcs.2015.04.001.
  17. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
  18. 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.
  19. Shivam Garg and Geevarghese Philip. Raising the bar for vertex cover: Fixed-parameter tractability above a higher guarantee. In Proceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1152-1166, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch80.
  20. Fanica Gavril. Testing for equality between maximum matching and minimum node covering. Information Processing Letters, 6(6):199-202, 1977. URL: https://doi.org/10.1016/0020-0190(77)90068-0.
  21. Wayne Goddard and Michael A. Henning. Independent domination in graphs: A survey and recent results. Discrete Mathematics, 313(7):839-854, 2013. URL: https://doi.org/10.1016/j.disc.2012.11.031.
  22. David G. Harris and N. S. Narayanaswamy. A faster algorithm for vertex cover parameterized by solution size, 2022. URL: http://arxiv.org/abs/2205.08022.
  23. Eva-Maria C. Hols, Stefan Kratsch, and Astrid Pieterse. Elimination distances, blocking sets, and kernels for vertex cover. In Proceedings of the 37th International Symposium on Theoretical Aspects of Computer Science (STACS), pages 36:1-36:14, 2020. URL: https://doi.org/10.4230/LIPIcs.STACS.2020.36.
  24. Yoichi Iwata and Yusuke Kobayashi. Improved analysis of highest-degree branching for feedback vertex set. Algorithmica, 83(8):2503-2520, 2021. URL: https://doi.org/10.1007/s00453-021-00815-w.
  25. Bart M. P. Jansen. The Power of Data Reduction: Kernels for Fundamental Graph Problems. PhD thesis, Utrecht University, 2013. URL: http://dspace.library.uu.nl/handle/1874/276438.
  26. Bart M. P. Jansen and Hans L. Bodlaender. Vertex cover kernelization revisited: Upper and lower bounds for a refined parameter. Theory of Computing Systems, 53(2):263-299, 2013. URL: https://doi.org/10.1007/s00224-012-9393-4.
  27. Subhash Khot and Venkatesh Raman. Parameterized complexity of finding subgraphs with hereditary properties. Theoretical Computer Science, 289(2):997-1008, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00414-5.
  28. Stefan Kratsch. A randomized polynomial kernelization for vertex cover with a smaller parameter. SIAM Journal on Discrete Mathematics, 32(3):1806-1839, 2018. URL: https://doi.org/10.1137/16M1104585.
  29. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. Journal of the ACM, 67(3):16:1-16:50, 2020. URL: https://doi.org/10.1145/3390887.
  30. M. Lesk, M. D. Plummer, and W. R. Pulleyblank. Equi-matchable graphs. Graph theory and combinatorics, pages 239-254, 1984. Google Scholar
  31. Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Transactions on Algorithms, 11(2):15:1-15:31, 2014. URL: https://doi.org/10.1145/2566616.
  32. László Lovász and Michael D. Plummer. Matching theory. North-Holland, 1986. Google Scholar
  33. Meena Mahajan and Venkatesh Raman. Parameterizing above guaranteed values: MaxSat and MaxCut. Journal of Algorithms, 31(2):335-354, 1999. URL: https://doi.org/10.1006/jagm.1998.0996.
  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. Michael D. Plummer. Some covering concepts in graphs. Journal of Combinatorial Theory, 8(1):91-98, 1970. URL: https://doi.org/10.1016/S0021-9800(70)80011-4.
  36. Michael D. Plummer. Well-covered graphs: A survey. Quaestiones Mathematicae, 16(3):253-287, 1993. URL: https://doi.org/10.1080/16073606.1993.9631737.
  37. Igor Razgon and Barry O'Sullivan. Almost 2-SAT is fixed-parameter tractable. Journal of Computer and System Sciences, 75(8):435-450, 2009. URL: https://doi.org/10.1016/j.jcss.2009.04.002.
  38. N. Robertson, P. Seymour, and R. Thomas. Quickly excluding a planar graph. Journal of Combinatorial Theory Series B, 62(2):323-348, 1994. URL: https://doi.org/10.1006/jctb.1994.1073.
  39. Neil Robertson and P. D. Seymour. Graph minors X: Obstructions to tree-decomposition. Journal of Combinatorial Theory Series B, 52(2):153-190, 1991. URL: https://doi.org/10.1016/0095-8956(91)90061-N.
  40. Ramesh S. Sankaranarayana and Lorna K. Stewart. Complexity results for well-covered graphs. Networks, 22(3):247-262, 1992. URL: https://doi.org/10.1002/net.3230220304.
  41. P. D. Seymour and Robin Thomas. Call routing and the ratcatcher. Combinatorica, 14(2):217-241, 1994. URL: https://doi.org/10.1007/BF01215352.
  42. Manuel Sorge and Mathias Weller. The graph parameter hierarchy. Unpublished manuscript, 2019. URL: https://manyu.pro/assets/parameter-hierarchy.pdf.
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