Exploiting c-Closure in Kernelization Algorithms for Graph Problems
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^đȘ(c), that Induced Matching admits a kernel with đȘ(câ· kâž) vertices, and that Irredundant Set admits a kernel with đȘ(c^{5/2} kÂł) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show.
Fixed-parameter tractability
kernelization
c-closure
Dominating Set
Induced Matching
Irredundant Set
Ramsey numbers
Theory of computation~Parameterized complexity and exact algorithms
Theory of computation~Graph algorithms analysis
65:1-65:17
Regular Paper
A continously updated version of the paper is available at https://arxiv.org/abs/2005.03986.
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).
Tomohiro
Koana
Tomohiro Koana
Technische UniversitĂ€t Berlin, Algorithmics and Computational Complexity, Germany
https://orcid.org/0000-0002-8684-0611
Supported by the Deutsche Forschungsgemeinschaft (DFG), project FPTinP, NI 369/19.
Christian
Komusiewicz
Christian Komusiewicz
Philipps-UniversitĂ€t Marburg, Fachbereich Mathematik und Informatik, Germany
https://orcid.org/0000-0003-0829-7032
Frank
Sommer
Frank Sommer
Philipps-UniversitĂ€t Marburg, Fachbereich Mathematik und Informatik, Germany
https://orcid.org/0000-0003-4034-525X
Supported by the Deutsche Forschungsgemeinschaft (DFG), project MAGZ, KO 3669/4-1.
10.4230/LIPIcs.ESA.2020.65
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Tomohiro Koana, Christian Komusiewicz, and Frank Sommer
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