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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2009.1843
URN: urn:nbn:de:0030-drops-18437
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2009/1843/
Fernau, Henning ;
Fomin, Fedor V. ;
Lokshtanov, Daniel ;
Raible, Daniel ;
Saurabh, Saket ;
Villanger, Yngve
Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves
Abstract
The {\sc $k$-Leaf Out-Branching} problem is to find an out-branching, that is a rooted oriented spanning tree, with at least $k$ leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the {\sc $k$-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted $k$-Leaf-Out-Branching}, a variant of {\sc $k$-Leaf-Out-Branching} where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics.
For the {\sc $k$-Leaf-Out-Branching} problem, we show that no polynomial kernel is possible unless the polynomial hierarchy collapses to third level by applying a recent breakthrough result by Bodlaender et al. (ICALP 2008) in a non-trivial fashion. However, our positive results for {\sc Rooted $k$-Leaf-Out-Branching} immediately imply that the seemingly intractable {\sc $k$-Leaf-Out-Branching} problem admits a data reduction to $n$ independent $O(k^3)$ kernels. These two results, tractability and intractability side by side, are the first ones separating {\it many-to-one kernelization} from {\it Turing kernelization}. This answers affirmatively an open problem regarding ``cheat kernelization'' raised by Mike Fellows and Jiong Guo independently.
BibTeX - Entry
@InProceedings{fernau_et_al:LIPIcs:2009:1843,
author = {Henning Fernau and Fedor V. Fomin and Daniel Lokshtanov and Daniel Raible and Saket Saurabh and Yngve Villanger},
title = {{Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves}},
booktitle = {26th International Symposium on Theoretical Aspects of Computer Science},
pages = {421--432},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-09-5},
ISSN = {1868-8969},
year = {2009},
volume = {3},
editor = {Susanne Albers and Jean-Yves Marion},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2009/1843},
URN = {urn:nbn:de:0030-drops-18437},
doi = {10.4230/LIPIcs.STACS.2009.1843},
annote = {Keywords: Parameterized algorithms, Kernelization, Out-branching, Max-leaf, Lower bounds}
}
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Keywords: |
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Parameterized algorithms, Kernelization, Out-branching, Max-leaf, Lower bounds |
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Collection: |
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26th International Symposium on Theoretical Aspects of Computer Science |
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Issue Date: |
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2009 |
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Date of publication: |
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19.02.2009 |
