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Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Chen, Jianer Fernau, Henning Kanj, Iyad A. Xia, Ge
Copyright Year	2007
Abstract	Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving $\mathcal{NP}$-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by $2k$, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by $335k$. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless $\mathcal{P} = \mathcal{NP}$, planar vertex cover does not have a problem kernel of size smaller than $4k/3$, and planar independent set and planar dominating set do not have kernels of size smaller than $2k$. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to $67 k$, improving significantly the $335 k$ previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.
Starting Page	1077
Ending Page	1106
Page Count	30
File Format	PDF
ISSN	00975397
DOI	10.1137/050646354
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	37
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2007-11-14
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems vertex cover kernel planar graph Graph algorithms dominating set parameterized algorithm independent set
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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