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R. Ganian, F. Slivovsky, St. Szeider:
"Meta-kernelization with Structural Parameters";
Talk: International Symposium on Mathematical Foundations of Computer Science (MFCS), Klosterneuburg, Austria; 2013-08-26 - 2013-08-30; in: "The 38th International Symposium on Mathematical Foundations of Computer Science, Proceedings", K. Chatterjee, J. Sgall (ed.); Springer / LNCS, 8087 (2013), ISBN: 978-3-642-40312-5; 457 - 468.

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 the first meta-kernelization theorems that use a structural parameters of the input and not the solution size.

Let C be a graph class. We define the C-cover number of a graph to be a the smallest number of modules the vertex set can be partitioned into, such that each module induces a subgraph that belongs to the class 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 C-cover number for any fixed class C of bounded rank-width (or equivalently, of bounded clique-width, or bounded Boolean width). Many graph problems such as Independent Dominating Set, c-Coloring, and c-Domatic Number 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.

http://dx.doi.org/10.1007/978-3-642-40313-2_41

Project Head Stefan Szeider:
The Parameterized Complexity of Reasoning Problems