Affiliation:
1. Universität Trier, Germany
2. University of Bergen, Bergen, Norway
3. The Institute of Mathematical Sciences, Chennai, India
Abstract
The
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
k
-Leaf-Out-Branching problem. We give the first polynomial kernel for Rooted
k
-Leaf-Out-Branching, a variant of
k
-Leaf-Out-Branching where the root of the tree searched for is also a part of the input. Our kernel with
O
(
k
3
) vertices is obtained using extremal combinatorics.
For the
k
-Leaf-Out-Branching problem, we show that no polynomial-sized kernel is possible unless
coNP
is in
NP/poly
. However, our positive results for Rooted
k
-Leaf-Out-Branching immediately imply that the seemingly intractable
k
-Leaf-Out-Branching problem admits a data reduction to
n
independent polynomial-sized kernels. These two results, tractability and intractability side by side, are the first ones separating
Karp kernelization
from
Turing kernelization
. This answers affirmatively an open problem regarding “cheat kernelization” raised by Mike Fellows and Jiong Guo independently.
Funder
Rigorous Theory of Preprocessing
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Reference36 articles.
1. Spanning Directed Trees with Many Leaves
2. Binkele-Raible D.
and
Fernau H
.
2012
. A faster exact algorithm for the directed maximum leaf spanning tree problem. In Computer Science in Russia CSR Lecture Notes in Computer Science vol.
6072
.
Springer 328--339. 10.1007/978-3-642-13182-0_31 Binkele-Raible D. and Fernau H. 2012. A faster exact algorithm for the directed maximum leaf spanning tree problem. In Computer Science in Russia CSR Lecture Notes in Computer Science vol. 6072. Springer 328--339. 10.1007/978-3-642-13182-0_31
3. Kernelization: New Upper and Lower Bound Techniques
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