Publication details

Kernelization Using Structural Parameters on Sparse Graph Classes

Authors

GAJARSKÝ Jakub HLINĚNÝ Petr OBDRŽÁLEK Jan ORDYNIAK Sebastian REIDL Felix ROSSMANITH Peter VILLAAMIL Fernando Sanchez SIKDAR Somnath

Year of publication 2013
Type Article in Proceedings
Conference ESA 2013
MU Faculty or unit

Faculty of Informatics

Citation
Doi http://dx.doi.org/10.1007/978-3-642-40450-4_45
Field Informatics
Keywords kernelization; parameterized complexity; sparse graphs
Description Meta-theorems for polynomial (linear) kernels have been the subject of intensive research in parameterized complexity. Heretofore, there were meta-theorems for linear kernels on graphs of bounded genus, H-minor-free graphs, and H-topological-minor-free graphs. To the best of our knowledge, there are no known meta-theorems for kernels for any of the larger sparse graph classes: graphs of bounded expansion, locally bounded expansion, and nowhere dense graphs. In this paper we prove meta-theorems for these three graph classes. More specifically, we show that graph problems that have finite integer index (FII) admit linear kernels on hereditary graphs of bounded expansion when parameterized by the size of a modulator to constant-treedepth graphs. For hereditary graph classes of locally bounded expansion, our result yields a quadratic kernel and for hereditary nowhere dense graphs, a polynomial kernel. While our parameter may seem rather strong, a linear kernel result on graphs of bounded expansion with a weaker parameter would for some problems violate known lower bounds. Moreover, we use a relaxed notion of FII which allows us to prove linear kernels for problems such as Longest Path/Cycle and Exact s,t-Path which do not have FII in general graphs.
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