Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves

Authors Henning Fernau, Fedor V. Fomin, Daniel Lokshtanov, Daniel Raible, Saket Saurabh, Yngve Villanger



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Henning Fernau
Fedor V. Fomin
Daniel Lokshtanov
Daniel Raible
Saket Saurabh
Yngve Villanger

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Henning Fernau, Fedor V. Fomin, Daniel Lokshtanov, Daniel Raible, Saket Saurabh, and Yngve Villanger. Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 421-432, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)
https://doi.org/10.4230/LIPIcs.STACS.2009.1843

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.
Keywords
  • Parameterized algorithms
  • Kernelization
  • Out-branching
  • Max-leaf
  • Lower bounds

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