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Exploiting c-Closure in Kernelization Algorithms for Graph Problems

Authors Tomohiro Koana , Christian Komusiewicz , Frank Sommer

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Fixed-parameter tractability
  • kernelization
  • c-closure
  • Dominating Set
  • Induced Matching
  • Irredundant Set
  • Ramsey numbers

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Abstract

A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number c such that G is c-closed. Fox et al. [SIAM J. Comput. '20] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^𝒪(c), that Induced Matching admits a kernel with 𝒪(c⁷ k⁸) vertices, and that Irredundant Set admits a kernel with 𝒪(c^{5/2} k³) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show.

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Tomohiro Koana, Christian Komusiewicz, and Frank Sommer. Exploiting c-Closure in Kernelization Algorithms for Graph Problems. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.65

Author Details

Tomohiro Koana
  • Technische Universität Berlin, Algorithmics and Computational Complexity, Germany
Christian Komusiewicz
  • Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany
Frank Sommer
  • Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany

Funding

  • Koana, Tomohiro: Supported by the Deutsche Forschungsgemeinschaft (DFG), project FPTinP, NI 369/19.
  • Sommer, Frank: Supported by the Deutsche Forschungsgemeinschaft (DFG), project MAGZ, KO 3669/4-1.

Acknowledgements

This work was started at the research retreat of the TU Berlin Algorithms and Computational Complexity group held in September 2019 at Schloss Neuhausen (Prignitz).

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