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Meta-kernelization with Structural Parameters

  • Robert Ganian
  • Friedrich Slivovsky
  • Stefan Szeider
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)

Abstract

Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS’09) and of Fomin et al. (FOCS’10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use structural parameters of the input and not the solution size. Let \(\mathcal{C}\) be a graph class. We define the \(\mathcal{C}\) - cover number of a graph to be the smallest number of modules the vertex set can be partitioned into such that each module induces a subgraph that belongs to the class \(\mathcal{C}\).

We show that each graph problem that can be expressed in Monadic Second Order (MSO) logic has a polynomial kernel with a linear number of vertices when parameterized by the \(\mathcal{C}\)- cover number for any fixed class \(\mathcal{C}\) of bounded rank-width (or equivalently, of bounded clique-width, or bounded Boolean-width). Many graph problems such as c -Coloring, c -Domatic Number and c -Clique Cover are covered by this meta-kernelization result.

Our second result applies to MSO expressible optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We show that these problems admit a polynomial annotated kernel with a linear number of vertices.

Keywords

Polynomial Time Vertex Cover Maximum Clique Input Graph Polynomial Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Robert Ganian
    • 1
  • Friedrich Slivovsky
    • 1
  • Stefan Szeider
    • 1
  1. 1.Institute of Information SystemsVienna University of TechnologyViennaAustria

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