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On problems without polynomial kernels

by Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, Danny Hermelin - LECT. NOTES COMPUT. SCI , 2007
"... Kernelization is a strong and widely-applied technique in parameterized complexity. In a nutshell, a kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parame ..."
Abstract - Cited by 143 (17 self) - Add to MetaCart
Kernelization is a strong and widely-applied technique in parameterized complexity. In a nutshell, a kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size

Subspace embeddings for the polynomial kernel

by Haim Avron, Huy L. Nguy˜̂en, David P. Woodruff - In NIPS , 2014
"... Sketching is a powerful dimensionality reduction tool for accelerating statistical learning algorithms. However, its applicability has been limited to a certain extent since the crucial ingredient, the so-called oblivious subspace embedding, can only be applied to data spaces with an explicit repres ..."
Abstract - Cited by 5 (3 self) - Add to MetaCart
without explicitly mapping the data to the high-dimensional space. In particular, we propose an embedding for mappings in-duced by the polynomial kernel. Using the subspace embeddings, we obtain the fastest known algorithms for computing an implicit low rank approximation of the higher-dimension mapping

A polynomial kernel for MULTICUT IN TREES

by Nicolas Bousquet, Jean Daligault, STÉPHAN THOMASSÉ , ANDERS YEO
"... The MULTICUT IN TREES problem consists in deciding, given a tree, a set of requests (i.e. paths in the tree) and an integer k, whether there exists a set of k edges cutting all the requests. This problem was shown to be FPT by Guo and Niedermeyer in [9]. They also provided an exponential kernel. T ..."
Abstract - Cited by 12 (2 self) - Add to MetaCart
. They asked whether this problem has a polynomial kernel. This question was also raised by Fellows in [1]. We show that MULTICUT IN TREES has a polynomial kernel.

A POLYNOMIAL KERNEL FOR MULTICUT IN TREES

by Anders Yeo, et al. , 2009
"... The MULTICUT IN TREES problem consists in deciding, given a tree, a set of requests (i.e. paths in the tree) and an integer k, whether there exists a set of k edges cutting all the requests. This problem was shown to be FPT by Guo and Niedermeyer in [10]. They also provided an exponential kernel. ..."
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. They asked whether this problem has a polynomial kernel. This question was also raised by Fellows in [1]. We show that MULTICUT IN TREES has a polynomial kernel.

Polynomial kernels collapse the W-hierarchy

by Liang Ding, Abdul Samad, Xingran Xue, Xiuzhen Huang, Liming Cai
"... Abstract. We prove that, for many parameterized problems in the class FPT, the existence of polynomial kernels implies the collapse of the W-hierarchy (i.e., W[P] = FPT). The collapsing results are also extended to assumed exponential kernels for problems in the class FPT. In partic-ular, we establ ..."
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Abstract. We prove that, for many parameterized problems in the class FPT, the existence of polynomial kernels implies the collapse of the W-hierarchy (i.e., W[P] = FPT). The collapsing results are also extended to assumed exponential kernels for problems in the class FPT. In partic-ular, we

A polynomial kernel for proper interval vertex deletion

by Fedor V. Fomin, Saket Saurabh, Yngve Villanger - Algorithms -ESA 2012, volume 7501 of Lecture Notes in Computer Science , 2012
"... Abstract. It is known that the problem of deleting at most k vertices to obtain a proper interval graph (Proper Interval Vertex Deletion) is fixed parameter tractable. However, whether the problem admits a polynomial kernel or not was open. Here, we answers this question in affirmative by obtaining ..."
Abstract - Cited by 1 (0 self) - Add to MetaCart
Abstract. It is known that the problem of deleting at most k vertices to obtain a proper interval graph (Proper Interval Vertex Deletion) is fixed parameter tractable. However, whether the problem admits a polynomial kernel or not was open. Here, we answers this question in affirmative by obtaining

Two edge modification problems without polynomial kernels

by Stefan Kratsch, Magnus Wahlström - IN IWPEC, VOLUME 5917 OF LNCS , 2009
"... Given a graph G and an integer k, the Π Edge Completion/Editing/Deletion problem asks whether it is possible to add, edit, or delete at most k edges in G such that one obtains a graph that fulfills the property Π. Edge modification problems have received considerable interest from a parameterized ..."
Abstract - Cited by 16 (2 self) - Add to MetaCart
point of view. When parameterized by k, many of these problems turned out to be fixed-parameter tractable and some are known to admit polynomial kernelizations, i.e. efficient preprocessing with a size guarantee that is polynomial in k. This paper answers an open problem posed by Cai (IWPEC 2006

Eigen-analysis of nonlinear PCA with polynomial kernels

by Zhiyu Liang, Yoonkyung Lee - Statistical Analysis and Data Mining
"... There has been growing interest in kernel methods for classification, clustering and dimension reduction. For example, kernel Fisher discriminant analysis, spectral clustering and kernel principal component analysis are widely used in statistical learning and data mining applications. The empirical ..."
Abstract - Cited by 1 (0 self) - Add to MetaCart
. In this paper, we examine the geometry of the nonlinear embedding for kernel PCA when polynomial kernels are used. We carry out eigen-analysis of the polynomial kernel operator associated with data distributions and investigate the effect of the degree of polynomial. The results provide both insights

Universal Consistency of Local Polynomial Kernel Regression Estimates

by Michael Kohler , 2000
"... Regression function estimation from independent and identically distributed data is considered. The L 2 error with integration with respect to the design measure is used as an error criterion. It is shown that suitably dened local polynomial kernel estimates are weakly and strongly universally consi ..."
Abstract - Cited by 2 (1 self) - Add to MetaCart
Regression function estimation from independent and identically distributed data is considered. The L 2 error with integration with respect to the design measure is used as an error criterion. It is shown that suitably dened local polynomial kernel estimates are weakly and strongly universally

On Polynomial Kernels for Sparse Integer Linear Programs

by Stefan Kratsch
"... Integer linear programs (ILPs) are a widely applied framework for dealing with combinatorial problems that arise in practice. It is known, e.g., by the success of CPLEX, that preprocessing and simplification can greatly speed up the process of optimizing an ILP. The present work seeks to further the ..."
Abstract - Cited by 1 (1 self) - Add to MetaCart
in time O(cn3 · mc ′). Thus, by a folklore argument, any such ILP admits a kernelization to an equivalent instance of size O(cn3). It is known, that unless NP ⊆ coNP/poly and the polynomial hierarchy collapses, no kernelization with size bound polynomial in n is possible. However, this lower bound only
Results 1 - 10 of 1,607