Document Open Access Logo

On the Parameterized Complexity of Clique Elimination Distance

Authors Akanksha Agrawal , M. S. Ramanujan

PDF

File

Thumbnail PDF
PDF
LIPIcs.IPEC.2020.1.pdf
  • Filesize: 0.6 MB
  • 13 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Elimination Distance
  • Cluster Graphs
  • Kernelization

Metrics

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

Abstract

Bulian and Dawar [Algorithmica, 2016] introduced the notion of elimination distance in an effort to define new tractable parameterizations for graph problems and showed that deciding whether a given graph has elimination distance at most k to any minor-closed class of graphs is fixed-parameter tractable parameterized by k [Algorithmica, 2017]. 
In this paper, we consider the problem of computing the elimination distance of a given graph to the class of cluster graphs and initiate the study of the parameterized complexity of a more general version - that of obtaining a modulator to such graphs. That is, we study the (η,Clq)-Elimination Deletion problem ((η,Clq)-ED Deletion) where, for a fixed η, one is given a graph G and k ∈ ℕ and the objective is to determine whether there is a set S ⊆ V(G) such that the graph G-S has elimination distance at most η to the class of cluster graphs. 
Our main result is a polynomial kernelization (parameterized by k) for this problem. As components in the proof of our main result, we develop a k^𝒪(η k + η²)n^𝒪(1)-time fixed-parameter algorithm for (η,Clq)-ED Deletion and a polynomial-time factor-min{𝒪(η⋅ opt⋅ log² n),opt^𝒪(1)} approximation algorithm for the same problem.

Cite As Get BibTex

Akanksha Agrawal and M. S. Ramanujan. On the Parameterized Complexity of Clique Elimination Distance. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 1:1-1:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.IPEC.2020.1

Author Details

Akanksha Agrawal
  • Indian Institute of Technology Madras, Chennai, India
M. S. Ramanujan
  • University of Warwick, Coventry, UK

Funding

  • Agrawal, Akanksha: supported by the PBC Program of Fellowships for Outstanding Post-doctoral Researchers from China and India when a part of the work was carried out.

References

  1. Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Polylogarithmic approximation algorithms for weighted-f-deletion problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 1:1-1:15, 2018. Google Scholar
  2. Marin Bougeret, Bart M. P. Jansen, and Ignasi Sau. Bridge-Depth Characterizes which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel. arXiv e-prints, 2020. URL: http://arxiv.org/abs/2004.12865.
  3. 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. Google Scholar
  4. Adam Bouland, Anuj Dawar, and Eryk Kopczynski. On tractable parameterizations of graph isomorphism. In Dimitrios M. Thilikos and Gerhard J. Woeginger, editors, Parameterized and Exact Computation - 7th International Symposium (IPEC), volume 7535, pages 218-230, 2012. Google Scholar
  5. Jannis Bulian and Anuj Dawar. Graph isomorphism parameterized by elimination distance to bounded degree. Algorithmica, 75(2):363-382, 2016. Google Scholar
  6. Jannis Bulian and Anuj Dawar. Fixed-parameter tractable distances to sparse graph classes. Algorithmica, 79(1):139-158, 2017. Google Scholar
  7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer Science & Business Media, 2015. Google Scholar
  8. Wojciech Czerwinski, Wojciech Nadara, and Marcin Pilipczuk. Improved bounds for the excluded-minor approximation of treedepth. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms (ESA), volume 144, pages 34:1-34:13, 2019. Google Scholar
  9. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  10. Eduard Eiben, Robert Ganian, Thekla Hamm, and O-joung Kwon. Measuring what matters: A hybrid approach to dynamic programming with treewidth. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS), volume 138, pages 42:1-42:15, 2019. Google Scholar
  11. Eduard Eiben, Robert Ganian, and Stefan Szeider. Meta-kernelization using well-structured modulators. In Thore Husfeldt and Iyad A. Kanj, editors, 10th International Symposium on Parameterized and Exact Computation (IPEC), volume 43 of LIPIcs, pages 114-126, 2015. Google Scholar
  12. Michael R. Fellows, Bart M. P. Jansen, and Frances A. Rosamond. Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity. Eur. J. Comb., 34(3):541-566, 2013. Google Scholar
  13. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, M. S. Ramanujan, and Saket Saurabh. Solving d-sat via backdoors to small treewidth. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 630-641, 2015. Google Scholar
  14. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar F-deletion: Approximation, kernelization and optimal FPT algorithms. In 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 470-479, 2012. Google Scholar
  15. Jakub Gajarský, Petr Hlinený, Jan Obdrzálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar. Kernelization using structural parameters on sparse graph classes. Journal of Computer and System Sciences, 84:219-242, 2017. Google Scholar
  16. Robert Ganian, Sebastian Ordyniak, and M. S. Ramanujan. Going beyond primal treewidth for (M)ILP. In Satinder P. Singh and Shaul Markovitch, editors, Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, pages 815-821, 2017. Google Scholar
  17. Robert Ganian, M. S. Ramanujan, and Stefan Szeider. Combining treewidth and backdoors for CSP. In Heribert Vollmer and Brigitte Vallée, editors, 34th Symposium on Theoretical Aspects of Computer Science (STACS), volume 66, pages 36:1-36:17, 2017. Google Scholar
  18. Archontia C. Giannopoulou, Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. Uniform kernelization complexity of hitting forbidden minors. ACM Transactions on Algorithms, 13(3):35:1-35:35, 2017. Google Scholar
  19. Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: Distance from triviality. In Rodney G. Downey, Michael R. Fellows, and Frank K. H. A. Dehne, editors, Parameterized and Exact Computation, First International Workshop (IWPEC), volume 3162, pages 162-173. Springer, 2004. Google Scholar
  20. Eva-Maria C. Hols, Stefan Kratsch, and Astrid Pieterse. Elimination distances, blocking sets, and kernels for vertex cover. In Christophe Paul and Markus Bläser, editors, 37th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 154, pages 36:1-36:14, 2020. Google Scholar
  21. Ashwin Jacob, Fahad Panolan, Venkatesh Raman, and Vibha Sahlot. Structural parameterizations with modulator oblivion, 2020. URL: http://arxiv.org/abs/2002.09972.
  22. Bart M. P. Jansen and Marcin Pilipczuk. Approximation and kernelization for chordal vertex deletion. SIAM J. Discrete Math., 32(3):2258-2301, 2018. Google Scholar
  23. David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. On generating all maximal independent sets. Information Processing Letters, 27(3):119-123, 1988. Google Scholar
  24. Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar. Linear kernels and single-exponential algorithms via protrusion decompositions. ACM Transactions on Algorithms, 12(2):21:1-21:41, 2016. Google Scholar
  25. Ton Kloks. Treewidth, Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer, 1994. Google Scholar
  26. Frank Thomson Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM, 46(6):787-832, 1999. Google Scholar
  27. Alexander Lindermayr, Sebastian Siebertz, and Alexandre Vigny. Elimination distance to bounded degree on planar graphs. In 45th International Symposium on Mathematical Foundations of Computer Science, MFCS, volume 170 of LIPIcs, pages 65:1-65:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  28. Kazuhisa Makino and Takeaki Uno. New algorithms for enumerating all maximal cliques. In 9th Scandinavian Workshop on Algorithm Theory (SWAT), pages 260-272, 2004. Google Scholar
  29. Bruce A. Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Operations Research Letters, 32(4):299-301, 2004. Google Scholar
  30. Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar. A faster parameterized algorithm for treedepth. In Automata, Languages, and Programming - 41st International Colloquium (ICALP), pages 931-942, 2014. Google Scholar
  31. Virginia Vassilevska Williams. Multiplying matrices faster than coppersmith-winograd. In Proceedings of the 44th Symposium on Theory of Computing Conference, (STOC), pages 887-898, 2012. Google Scholar
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact