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Approximate Turing Kernelization and Lower Bounds for Domination Problems

Authors Stefan Kratsch , Pascal Kunz

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Approximate Turing kernelization
  • approximation lower bounds
  • exponential-time hypothesis
  • dominating set
  • capacitated dominating
  • connected dominating set
  • independent dominating set
  • treewidth
  • vertex cover number

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Abstract

An α-approximate polynomial Turing kernelization is a polynomial-time algorithm that computes an (α c)-approximate solution for a parameterized optimization problem when given access to an oracle that can compute c-approximate solutions to instances with size bounded by a polynomial in the parameter. Hols et al. [ESA 2020] showed that a wide array of graph problems admit a (1+ε)-approximate polynomial Turing kernelization when parameterized by the treewidth of the graph and left open whether Dominating Set also admits such a kernelization.
We show that Dominating Set and several related problems parameterized by treewidth do not admit constant-factor approximate polynomial Turing kernelizations, even with respect to the much larger parameter vertex cover number, under certain reasonable complexity assumptions. On the positive side, we show that all of them do have a (1+ε)-approximate polynomial Turing kernelization for every ε > 0 for the joint parameterization by treewidth and maximum degree, a parameter which generalizes cutwidth, for example.

Cite As Get BibTex

Stefan Kratsch and Pascal Kunz. Approximate Turing Kernelization and Lower Bounds for Domination Problems. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.IPEC.2023.32

Author Details

Stefan Kratsch
  • Algorithm Engineering, Humboldt-Universität zu Berlin, Germany
Pascal Kunz
  • Algorithm Engineering, Humboldt-Universität zu Berlin, Germany

Funding

  • Kunz, Pascal: Supported by the DFG Research Training Group 2434 "Facets of Complexity".

References

  1. Daniel Binkele-Raible, Henning Fernau, Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Yngve Villanger. Kernel(s) for problems with no kernel: On out-trees with many leaves. ACM Transactions on Algorithms, 8(4), 2012. URL: https://doi.org/10.1145/2344422.2344428.
  2. M. Chlebík and J. Chlebíková. Approximation hardness of dominating set problems in bounded degree graphs. Information and Computation, 206(11):1264-1275, 2008. URL: https://doi.org/10.1016/j.ic.2008.07.003.
  3. Vasek Chvátal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233-235, 1979. URL: https://doi.org/10.1287/moor.4.3.233.
  4. Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization lower bounds through colors and IDs. ACM Transactions on Algorithms, 11(2):1-20, 2014. URL: https://doi.org/10.1145/2650261.
  5. Uriel Feige, MohammadTaghi Hajiaghayi, and James R. Lee. Improved approximation algorithms for minimum weight vertex separators. SIAM Journal on Computing, 38(2):629-657, 2008. URL: https://doi.org/10.1137/05064299X.
  6. Michael R. Fellows, Ariel Kulik, Frances Rosamond, and Hadas Shachnai. Parameterized approximation via fidelity preserving transformations. Journal of Computer and System Sciences, 93:30-40, 2018. URL: https://doi.org/10.1016/j.jcss.2017.11.001.
  7. 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.
  8. Sudipto Guha and Samir Khuller. Approximation algorithms for connected dominating sets. Algorithmica, 20:374-387, 1998. URL: https://doi.org/10.1007/PL00009201.
  9. Magnús M. Halldórsson. Approximating the minimum maximal independence number. Information Processing Letters, 46(4):169-172, 1993. URL: https://doi.org/10.1016/0020-0190(93)90022-2.
  10. Pinar Heggernes, Pim van ’t Hof, Bart M.P. Jansen, Stefan Kratsch, and Yngve Villanger. Parameterized complexity of vertex deletion into perfect graph classes. Theoretical Computer Science, 511:172-180, 2013. URL: https://doi.org/10.1016/j.tcs.2012.03.013.
  11. Danny Hermelin, Stefan Kratsch, Karolina Sołtys, Magnus Wahlström, and Xi Wu. A completeness theory for polynomial (Turing) kernelization. Algorithmica, 71(3):702-730, 2015. URL: https://doi.org/10.1007/s00453-014-9910-8.
  12. Eva-Maria C. Hols, Stefan Kratsch, and Astrid Pieterse. Approximate Turing kernelization for problems parameterized by treewidth. In Proceedings of the 28th Annual European Symposium on Algorithms (ESA), pages 60:1-60:23, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.60.
  13. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  14. Robert W. Irving. On approximating the minimum independent dominating set. Information Processing Letters, 37(4):197-200, 1991. URL: https://doi.org/10.1016/0020-0190(91)90188-N.
  15. Bart M. P. Jansen. Turing kernelization for finding long paths and cycles in restricted graph classes. Journal of Computer and System Sciences, 85:18-37, 2017. URL: https://doi.org/10.1016/j.jcss.2016.10.008.
  16. Bart M. P. Jansen, Marcin Pilipczuk, and Marcin Wrochna. Turing kernelization for finding long paths in graphs excluding a topological minor. In Proceedings of the 12th International Symposium on Parameterized and Exact Computation (IPEC), pages 23:1-23:13, 2018. URL: https://doi.org/10.4230/LIPIcs.IPEC.2017.23.
  17. Philip Klein and R. Ravi. A nearly best-possible approximation algorithm for node-weighted Steiner trees. Journal of Algorithms, 19(1):104-115, 1995. URL: https://doi.org/10.1006/jagm.1995.1029.
  18. Ton Kloks. Treewidth: Computations and Approximations. Springer, 1994. URL: https://doi.org/10.1007/BFb0045375.
  19. Daniel Lokshtanov. Wheel-free deletion is W[2]-hard. In Proceedings of the 3rd International Symposium on Parameterized and Exact Computation (IPEC), pages 141-147, 2008. URL: https://doi.org/10.1007/978-3-540-79723-4_14.
  20. Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. Lossy kernelization. CoRR, abs/1604.04111, 2016. Full version of [Daniel Lokshtanov et al., 2017]. URL: https://doi.org/10.48550/arXiv.1604.04111.
  21. Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. Lossy kernelization. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC 2017), pages 224-237, 2017. URL: https://doi.org/10.1145/3055399.3055456.
  22. Jelani Nelson. A note on set cover inapproximability independent of universe size. Electronic Colloquium on Computational Complexity, TR07-105, 2007. URL: https://eccc.weizmann.ac.il/eccc-reports/2007/TR07-105/index.html.
  23. Laurence A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2(4):385-393, 1982. URL: https://doi.org/10.1007/BF02579435.
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