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
Reads0
Chats0
TLDR
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
Citations
More filters
Proceedings ArticleDOI
Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels
Bart M. P. Jansen,Dániel Marx +1 more
TL;DR: It is shown that if every graph of a hereditary class F satisfies the property that it is possible to delete a bounded number of vertices such that every remaining component has size at most two, then F-Subgraph Test is solvable in randomized polynomial time and it is NP-hard otherwise.
Book ChapterDOI
Finding Highly Connected Subgraphs
TL;DR: This work examines the computational complexity of finding highly connected subgraphs, and finds that among the parameters yielding tractability, the edge isolation seems to provide the best trade-off between running time bounds and a small parameter value in relevant instances.
Journal ArticleDOI
Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs
TL;DR: Two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs are developed based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem-specific combinatorial structures, and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs.
Journal ArticleDOI
Dynamic Dominating Set and Turbo-Charging Greedy Heuristics
TL;DR: In this paper, a dynamic version of the DOMINATING SET problem is introduced and proved to be fixed-parameter tractable (FPT) in settings where problem instances evolve, and the main purpose of this paper is to exposit two very different but very general, motivational schemes in the art of parameterization.
Book ChapterDOI
A Completeness Theory for Polynomial (Turing) Kernelization
TL;DR: This work defines two kernelization hardness hierarchies which are akin to the M- and W-hierarchies of ordinary parameterized complexity, and conjecture that no WK[1]-hard problem admits a polynomial Turing kernel.
References
More filters
Reducibility Among Combinatorial Problems.
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.
Book
Parameterized Complexity
TL;DR: An approach to complexity theory which offers a means of analysing algorithms in terms of their tractability, and introduces readers to new classes of algorithms which may be analysed more precisely than was the case until now.
Book
Parameterized complexity theory
Jörg Flum,Martin Grohe +1 more
TL;DR: Fixed-Parameter Tractability.
Book
Invitation to fixed-parameter algorithms
TL;DR: This paper discusses Fixed-Parameter Algorithms, Parameterized Complexity Theory, and Selected Case Studies, and some of the techniques used in this work.