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Linear-time Kernelization for Feedback Vertex Set [article]

Yoichi Iwata
2017 arXiv   pre-print
This is the first linear-time polynomial-size kernel for Feedback Vertex Set.  ...  We note that under the assumption of NP⊆coNP/poly, Feedback Vertex Set does not admit an O(k^2-ϵ)-size kernel for any ϵ>0.  ...  Our Contribution In this paper, we propose a linear-time quadratic-size kernel for Feedback Vertex Set. This is the first linear-time polynomial-size kernel for this problem.  ... 
arXiv:1608.01463v3 fatcat:qg7pomkxnzbyve4tsexgyq53uy

Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

Bart M. P. Jansen, Hans L. Bodlaender, Marc Herbstritt
2011 Symposium on Theoretical Aspects of Computer Science  
Vertex Cover using a "refined" parameter: Feedback Vertex Number We have studied data reduction for Vertex Cover using a "refined" parameter: Feedback Vertex Number  Kernel with O(|X| 3 ) vertices  Usage  ...  Cover using a "refined" parameter: Feedback Vertex Number  Kernel with O(|X| 3 ) vertices  Usage of vertex weights affects kernelizability  No polynomial kernel for weighted problem parameterized by  ... 
doi:10.4230/lipics.stacs.2011.177 dblp:conf/stacs/JansenB11 fatcat:v6mfndllc5bsxctpwhranx52wa

The Power of Linear-Time Data Reduction for Maximum Matching

George B. Mertzios, André Nichterlein, Rolf Niedermeier
2020 Algorithmica  
As a warm-up we first show that a subset of our data reduction rules for the "feedback vertex set kernel" also yields a linear-time computable linear-size kernel for the typically much larger parameter  ...  Notably, there is a trivial linear-time algorithm for computing the feedback edge number and there is a linear-time factor-4 approximation algorithm for the feedback vertex number [1] .  ...  Warm-Up: Parameter Feedback Edge Number We provide a linear-time computable linear-size kernel for Matching parameterized by the feedback edge number, that is, the size of a minimum feedback edge set.  ... 
doi:10.1007/s00453-020-00736-0 fatcat:jcfxcsmjsrhftb6ieutarcmjeq

The Undirected Feedback Vertex Set Problem Has a Poly(k) Kernel [chapter]

Kevin Burrage, Vladimir Estivill-Castro, Michael Fellows, Michael Langston, Shev Mac, Frances Rosamond
2006 Lecture Notes in Computer Science  
Resolving a noted open problem, we show that the Undirected Feedback Vertex Set problem, parameterized by the size of the solution set of vertices, is in the parameterized complexity class P oly(k), that  ...  Our main result shows an O(k 11 ) kernelization bound.  ...  It makes sense to ask if the Undirected Feedback Vertex Set problem might admit a linear-size Turing kernelization.  ... 
doi:10.1007/11847250_18 fatcat:gjnuswvyvbb2jlsnrxptg2cdga

Kernels for Feedback Arc Set In Tournaments [article]

Stéphane Bessy, Fedor V. Fomin, Serge Gaspers, Christophe Paul, Anthony Perez, Saket Saurabh, Stéphan Thomassé
2009 arXiv   pre-print
The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST.  ...  Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST.  ...  In Section 2, we give some definition and preliminary results regarding feedback arc sets. In Section 3 we give a linear vertex kernel for k-FAST.  ... 
arXiv:0907.2165v2 fatcat:xdjh2kzqdjdeldxl2dwr7riaci

Kernels for feedback arc set in tournaments

Stéphane Bessy, Fedor V. Fomin, Serge Gaspers, Christophe Paul, Anthony Perez, Saket Saurabh, Stéphan Thomassé
2011 Journal of computer and system sciences (Print)  
The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST.  ...  Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST. *  ...  In Section 2, we give some definition and preliminary results regarding feedback arc sets. In Section 3 we give a linear vertex kernel for k-FAST.  ... 
doi:10.1016/j.jcss.2010.10.001 fatcat:6hmv67xnzvgyjma2y4loaszvvq

Fixed-parameter tractability results for full-degree spanning tree and its dual

Jiong Guo, Rolf Niedermeier, Sebastian Wernicke
2009 Networks  
We provide first-time fixed-parameter tractability results for the NP-complete problems Maximum Full-Degree Spanning Tree and Minimum-Vertex Feedback Edge Set.  ...  For Minimum-Vertex Feedback Edge Set the task is to minimize the number of vertices that end up with a reduced degree.  ...  [1] for proving a linear-size kernel for Dominating Set in planar graphs.  ... 
doi:10.1002/net.20353 fatcat:3dufkge6abgxvjkpewufyipqg4

Fixed-Parameter Tractability Results for Full-Degree Spanning Tree and Its Dual [chapter]

Jiong Guo, Rolf Niedermeier, Sebastian Wernicke
2006 Lecture Notes in Computer Science  
We provide first-time fixed-parameter tractability results for the NP-complete problems Maximum Full-Degree Spanning Tree and Minimum-Vertex Feedback Edge Set.  ...  For Minimum-Vertex Feedback Edge Set the task is to minimize the number of vertices that end up with a reduced degree.  ...  [1] for proving a linear-size kernel for Dominating Set in planar graphs.  ... 
doi:10.1007/11847250_19 fatcat:tvodrx6z5beb7kkwvuxipjl6fe

New Races in Parameterized Algorithmics [chapter]

Christian Komusiewicz, Rolf Niedermeier
2012 Lecture Notes in Computer Science  
Herein, the attention usually focuses on improving the running time factor exponential in the considered parameter, and, in case of kernelization algorithms, to improve the bound on the kernel size.  ...  We discuss several of these aspects and particularly focus on the search for "stronger parameterizations" in developing fixed-parameter algorithms.  ...  We thank André Nichterlein, Manuel Sorge, and Mathias Weller for their comments which have improved this article.  ... 
doi:10.1007/978-3-642-32589-2_2 fatcat:gpfsek5febh5rikaagecihqclq

Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs [chapter]

Sepp Hartung, Christian Komusiewicz, André Nichterlein
2012 Lecture Notes in Computer Science  
On the positive side, we give polynomial-size problem kernels for the parameters feedback edge set size of G and size of a cluster editing set of G and present a direct combinatorial algorithm for the  ...  Given an undirected graph G = (V, E) and an integer ≥ 1, the NPhard 2-Club problem asks for a vertex set S ⊆ V of size at least such that the subgraph induced by S has diameter at most two.  ...  A Linear Kernel for the Parameter Feedback Edge Set Size A feedback edge set F of a graph G is an edge set whose deletion transforms the graph G into a forest.  ... 
doi:10.1007/978-3-642-33293-7_22 fatcat:5hbgpg6wuzbcbetfwpivstuxle

Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs

Sepp Hartung, Christian Komusiewicz, André Nichterlein
2015 Journal of Graph Algorithms and Applications  
On the positive side, we give polynomial-size problem kernels for the parameters feedback edge set size of G and size of a cluster editing set of G and present a direct combinatorial algorithm for the  ...  Given an undirected graph G = (V, E) and an integer ≥ 1, the NPhard 2-Club problem asks for a vertex set S ⊆ V of size at least such that the subgraph induced by S has diameter at most two.  ...  A Linear Kernel for the Parameter Feedback Edge Set Size A feedback edge set F of a graph G is an edge set whose deletion transforms the graph G into a forest.  ... 
doi:10.7155/jgaa.00352 fatcat:5xnvb6pycbf2rjuz35ju7p7tgq

On tractable cases of Target Set Selection

André Nichterlein, Rolf Niedermeier, Johannes Uhlmann, Mathias Weller
2012 Social Network Analysis and Mining  
number", "vertex cover number", and "feedback edge set number" of the underlying graph on the problem's complexity, revealing both tractable and intractable cases.  ...  TSS can be efficiently solved on graphs with small feedback edge set number and also turns out to be fixed-parameter tractable when parameterized by the vertex cover number, both results contrasting known  ...  TARGET SET SELECTION admits a problem kernel of size O(f ), where f denotes the feedback edge set number. The kernelization runs in linear time.  ... 
doi:10.1007/s13278-012-0067-7 fatcat:k2q4nk4qnbatnemcvpvm32tqjq

On Tractable Cases of Target Set Selection [chapter]

André Nichterlein, Rolf Niedermeier, Johannes Uhlmann, Mathias Weller
2010 Lecture Notes in Computer Science  
number", "vertex cover number", and "feedback edge set number" of the underlying graph on the problem's complexity, revealing both tractable and intractable cases.  ...  TSS can be efficiently solved on graphs with small feedback edge set number and also turns out to be fixed-parameter tractable when parameterized by the vertex cover number, both results contrasting known  ...  TARGET SET SELECTION admits a problem kernel of size O(f ), where f denotes the feedback edge set number. The kernelization runs in linear time.  ... 
doi:10.1007/978-3-642-17517-6_34 fatcat:utuzuybyojah5iljtjbk3j2cuq

The Power of Data Reduction for Matching [article]

George B. Mertzios, André Nichterlein, Rolf Niedermeier
2017 arXiv   pre-print
We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on (almost) linear-time kernelization.  ...  For m-edge and n-vertex graphs, it is well-known to be solvable in O(m√(n)) time; however, for several applications this running time is still too slow.  ...  As a warm-up we first show that a subset of our data reduction rules for the "feedback vertex set kernel" also yields a linear-time computable linear-size kernel for the typically much larger parameter  ... 
arXiv:1609.08879v2 fatcat:7kxmo2z3nnehlpa5tngr3ziyzy

Dynamic Parameterized Problems

R. Krithika, Abhishek Sahu, Prafullkumar Tale
2017 Algorithmica  
For specific cases of Dynamic Π-Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved FPT algorithms and give linear kernels.  ...  Then, we show that Dynamic Feedback Vertex Set admits a randomized algorithm with 1.6667 k n O(1) running time.  ...  We are also thankful to the reviewers for their suggestions that strengthened the results for Dynamic Π-Deletion.  ... 
doi:10.1007/s00453-017-0349-6 fatcat:w42snepu6zc3helts3t4xyum6a
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