Kernelization through Tidying

Bevern, René van; Moser, Hannes; Niedermeier, Rolf

doi:10.1007/978-3-642-12200-2_46

Cited by 3 publications

(2 citation statements)

References 9 publications

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“…However, in a method for kernelizing vertex deletion problems whose goal graphs can be characterized by forbidden induced subgraphs, van Bevern et al [27] show how polynomial time approximation results can be exploited in kernelization.…”

Section: Related Workmentioning

confidence: 99%

Parameterized Approximation via Fidelity Preserving Transformations

Fellows

Kulik

Rosamond

et al. 2012

Automata, Languages, and Programming

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Abstract.We motivate and describe a new parameterized approximation paradigm which studies the interaction between performance ratio and running time for any parameterization of a given optimization problem. As a key tool, we introduce the concept of α-shrinking transformation, for α ≥ 1. Applying such transformation to a parameterized problem instance decreases the parameter value, while preserving approximation ratio of α (or α-fidelity). For example, it is well-known that Vertex Cover cannot be approximated within any constant factor better than 2 [22] (under usual assumptions). Our parameterized α-approximation algorithm for k-Vertex Cover, parameterized by the solution size, has a running time of 1.273(2−α)k , where the running time of the best FPT algorithm is 1.273 k [10]. Our algorithms define a continuous tradeoff between running times and approximation ratios, allowing practitioners to appropriately allocate computational resources. Moving even beyond the performance ratio, we call for a new type of approximative kernelization race. Our α-shrinking transformations can be used to obtain kernels which are smaller than the best known for a given problem. For the Vertex Cover problem we obtain a kernel size of 2(2 − α)k. The smaller "α-fidelity" kernels allow us to solve exactly problem instances more efficiently, while obtaining an approximate solution for the original instance. We show that such transformations exist for several fundamental problems, including Vertex Cover, d-Hitting Set, Connected Vertex Cover and Steiner Tree. We note that most of our algorithms are easy to implement and are therefore practical in use.

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Section: Related Workmentioning

confidence: 99%

Parameterized Approximation via Fidelity Preserving Transformations

Fellows

Kulik

Rosamond

et al. 2012

Automata, Languages, and Programming

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“…Related results. While a lot of kernelization results are known for graph modification problems [5,15,18,19], fewer results exist regarding directed graph and hypergraph modification problems. An example of polynomial kernel for a directed graph modification problem is the quadratic vertex-kernel for Transitivity Editing [20].…”

Section: Introductionmentioning

confidence: 99%

Linear Vertex-kernels for Several Dense Ranking r -Constraint Satisfaction Problems

Perez

2013

Lecture Notes in Computer Science

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A ranking r-constraint satisfaction problem (ranking r-CSP for short) consists of a ground set of vertices V , an arity r 2, a parameter k ∈ N and a constraint system c, where c is a function which maps rankings of r-sized sets S ⊆ V to {0, 1}. The objective is to decide if there exists a ranking σ of the vertices satisfying all but at most k constraints (i.e. S⊆V,|S|=r c(σ(S)) k). Famous ranking r-CSPs include Feedback Arc Set in Tournaments and Dense Betweenness [4,13]. We consider such problems from the kernelization viewpoint. We prove that so-called l r -simply characterized ranking r-CSPs admit linear vertex-kernels whenever they admit constant-factor approximation algorithms. This implies that r-Dense Betweenness and r-Dense Transitive Feedback Arc Set [13], two natural generalizations of the previously mentioned problems, admit linear vertex-kernels. Moreover, we introduce another generalization of Feedback Arc Set in Tournaments, which does not fit the aforementioned framework. Based on techniques from [10] we obtain a 5-approximation, and then provide a linear vertex-kernel.

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An overview of kernelization algorithms for graph modification problems

Liu

Wang

Guo

2014

Tinshhua Sci. Technol.

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Kernelization algorithms for graph modification problems are important ingredients in parameterized computation theory. In this paper, we survey the kernelization algorithms for four types of graph modification problems, which include vertex deletion problems, edge editing problems, edge deletion problems, and edge completion problems. For each type of problem, we outline typical examples together with recent results, analyze the main techniques, and provide some suggestions for future research in this field.

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Kernelization through Tidying

Cited by 3 publications

References 9 publications

Parameterized Approximation via Fidelity Preserving Transformations

Parameterized Approximation via Fidelity Preserving Transformations

Linear Vertex-kernels for Several Dense Ranking r -Constraint Satisfaction Problems

An overview of kernelization algorithms for graph modification problems

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