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.

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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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