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27
PolynomialTime Data Reduction for DOMINATING SET
 Journal of the ACM
, 2004
"... Dealing with the NPcomplete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a socalled problem kernel of linear size, achiev ..."
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Cited by 65 (8 self)
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Dealing with the NPcomplete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a socalled problem kernel of linear size, achieved by two simple and easy to implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization.
The bidimensionality Theory and Its Algorithmic Applications
 Computer Journal
, 2005
"... This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixedparameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, boundedgenus graphs and gra ..."
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Cited by 49 (3 self)
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This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixedparameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, boundedgenus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the Graph Minor Theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. Here, we summarize the known combinatorial and algorithmic results of bidimensionality theory with the highlevel ideas involved in their proof; we describe the previous work on which the theory is based and/or extends; and we mention several remaining open problems. 1.
Lower bounds based on the Exponential Time Hypothesis
 Bulletin of the EATCS
, 2011
"... In this article we survey algorithmic lower bound results that have been obtained in the field of exact exponential time algorithms and parameterized complexity under certain assumptions on the running time of algorithms solving CNFSat, namely Exponential time hypothesis (ETH) and Strong Exponenti ..."
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Cited by 35 (3 self)
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In this article we survey algorithmic lower bound results that have been obtained in the field of exact exponential time algorithms and parameterized complexity under certain assumptions on the running time of algorithms solving CNFSat, namely Exponential time hypothesis (ETH) and Strong Exponential time hypothesis (SETH). 1
Fixedparameter algorithms for the (k, r)center in planar graphs and map graphs
 ACM TRANSACTIONS ON ALGORITHMS
, 2003
"... ..."
On the optimality of planar and geometric approximation schemes
"... We show for several planar and geometric problems that the best known approximation schemes are essentially optimal with respect to the dependence on ǫ. For example, we show that the 2O(1/ǫ) · n time approximation schemes for planar MAXIMUM INDEPENDENT SET and for TSP on a metric defined by a plan ..."
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Cited by 17 (5 self)
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We show for several planar and geometric problems that the best known approximation schemes are essentially optimal with respect to the dependence on ǫ. For example, we show that the 2O(1/ǫ) · n time approximation schemes for planar MAXIMUM INDEPENDENT SET and for TSP on a metric defined by a planar graph are essentially optimal: if there is a δ> 0 such that any of these problems admits a 2O((1/ǫ)1−δ) O(1) n time PTAS, then the Exponential Time Hypothesis (ETH) fails. It is known that MAXIMUM INDEPENDENT SET on unit disk graphs and the planar logic problems MPSAT, TMIN, TMAX admit nO(1/ǫ) time approximation schemes. We show that they are optimal in the sense that if there is a δ> 0 such that any of these problems admits a 2 (1/ǫ)O(1) nO((1/ǫ)1−δ) time PTAS, then ETH fails.
Parameterized Complexity of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 15 (5 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter intractability results are surveyed as well. Finally, we give some directions for future research.
Polynomial time data reduction for Dominating Set
 Journal of the ACM
"... Dealing with the NPcomplete Dominating Set problem on undirected graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a socalled problem kernel of linear size, ..."
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Cited by 13 (7 self)
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Dealing with the NPcomplete Dominating Set problem on undirected graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a socalled problem kernel of linear size, achieved by two simple and easy to implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization. 1
Theory and application of width bounded geometric separator
 In 23 rd Annual Symposium on Theoretical Aspects of Computer Science (STACS
, 2006
"... Abstract. We introduce the notion of the width bounded geometric separator and develop the techniques for the existence of the width bounded separator in any ddimensional Euclidean space. The separator is applied in obtaining 2 O( √ n) time exact algorithms for a class of NPcomplete geometric probl ..."
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Cited by 7 (2 self)
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Abstract. We introduce the notion of the width bounded geometric separator and develop the techniques for the existence of the width bounded separator in any ddimensional Euclidean space. The separator is applied in obtaining 2 O( √ n) time exact algorithms for a class of NPcomplete geometric problems, whose previous algorithms take n O( √ n) time [2,5,1]. One of those problems is the well known disk covering problem, which seeks to determine the minimal number of fixed size disks to cover n points on a plane [10]. They also include some NPhard problems on disk graphs such as the maximum independent set problem, the vertex cover problem, and the minimum dominating set problem. 1
Optimization problems in unitdisk graphs
 In Christodoulos A. Floudas and Panos M. Pardalos, editors, Encyclopedia of Optimization
, 2009
"... UnitDisk Graphs (UDGs) are intersection graphs of equal diameter (or unit diameter w.l.o.g.) circles in the Euclidean plane. In the geometric (or disk) representation, each ..."
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Cited by 6 (0 self)
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UnitDisk Graphs (UDGs) are intersection graphs of equal diameter (or unit diameter w.l.o.g.) circles in the Euclidean plane. In the geometric (or disk) representation, each
Bidimensionality and Geometric Graphs
"... Bidimensionality theory was introduced by Demaine et al. [JACM 2005] as a framework to obtain algorithmic results for hard problems on minor closed graph classes. The theory has been sucessfully applied to yield subexponential time parameterized algorithms, EPTASs and linear kernels for many problem ..."
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Cited by 5 (3 self)
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Bidimensionality theory was introduced by Demaine et al. [JACM 2005] as a framework to obtain algorithmic results for hard problems on minor closed graph classes. The theory has been sucessfully applied to yield subexponential time parameterized algorithms, EPTASs and linear kernels for many problems on families of graphs excluding a fixed graph H as a minor. In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for FEEDBACK VERTEX SET, VERTEX COVER, CONNECTED VERTEX COVER, DIAMOND HITTING SET, on map graphs and unit disk graphs, and for CYCLE PACKING and MINIMUMVERTEX FEEDBACK EDGE SET on unit disk graphs. To the best of our knowledge, these results were previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for VERTEX COVER, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. Our results are based on the recent decomposition theorems proved by Fomin et al. in [SODA