We study a general class of problems called F-Deletion problems. In an F-Deletion problem, we are asked whether a subset of at most k vertices can be deleted from a graph G such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-Deletion problem when F contains a planar graph. We give - a linear vertex kernel on graphs excluding t-claw K_(1,t), the star with t leaves, as an induced subgraph, where t is a fixed integer. - an approximation algorithm achieving an approximation ratio of O(log^(3/2) OPT), where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F only contains graph theta_c as a minor for a fixed integer c. The graph theta_c consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.
@InProceedings{fomin_et_al:LIPIcs.STACS.2011.189, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Misra, Neeldhara and Philip, Geevarghese and Saurabh, Saket}, title = {{Hitting forbidden minors: Approximation and Kernelization}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {189--200}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.189}, URN = {urn:nbn:de:0030-drops-30103}, doi = {10.4230/LIPIcs.STACS.2011.189}, annote = {Keywords: kernelization} }
Feedback for Dagstuhl Publishing