Journal ArticleDOI
Kernel(s) for problems with no kernel: On out-trees with many leaves
Daniel Binkele-Raible,Henning Fernau,Fedor V. Fomin,Daniel Lokshtanov,Saket Saurabh,Yngve Villanger +5 more
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For the k-Leaf-Out-Branching problem, it was shown in this paper that no polynomial-sized kernel is possible unless coNP is in NP/poly.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(k3) 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.read more
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Journal ArticleDOI
On some FPT problems without polynomial Turing compressions
TL;DR: In this article , it was shown that edge clique cover, integer linear programming, and a -choosability parameterized by some structural parameters are NP-hard under Cook reductions even if their parameter values are small.
Posted Content
How heavy independent sets help to find arborescences with many leaves in DAGs
TL;DR: In this paper, a 7/5-approximation algorithm for the maximum leaf spanning arborescence problem on rooted directed acyclic graphs was proposed, which is the best known algorithm for this problem.
Journal ArticleDOI
Introducing lop-Kernels: A Framework for Kernelization Lower Bounds
TL;DR: In this paper , it was shown that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. (2015) , and that the known cubic kernel for maximum minimal feedback vertex set is also essentially optimal.
References
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TL;DR: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.
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Jörg Flum,Martin Grohe +1 more
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TL;DR: This paper discusses Fixed-Parameter Algorithms, Parameterized Complexity Theory, and Selected Case Studies, and some of the techniques used in this work.