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Linear kernels and singleexponential algorithms via protrusion decompositions
, 2012
"... A ttreewidthmodulator of a graph G is a set X ⊆ V (G) such that the treewidth of G−X is at most t − 1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a ttreewidthmodulator. Similar decompositions have already been explicitly or ..."
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Cited by 16 (5 self)
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A ttreewidthmodulator of a graph G is a set X ⊆ V (G) such that the treewidth of G−X is at most t − 1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a ttreewidthmodulator. Similar decompositions have already been explicitly or implicitly used for obtaining polynomial kernels [3, 7, 33, 43]. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and is treewidthbounding admits a linear kernel on the class of Htopologicalminorfree graphs, where H is some arbitrary but fixed graph. A parameterized graph problem is called treewidthbounding if all positive instances have a ttreewidthmodulator of size O(k), for some constant t. This result partially extends previous metatheorems on the existence of linear kernels on graphs of bounded genus [7] and Hminorfree graphs [37]. In particular, we show that Chordal Vertex Deletion, Interval Vertex Deletion, Treewidtht Vertex Deletion, and Edge Dominating Set have linear kernels on Htopologicalminorfree graphs.
Linear kernels for (connected) dominating set on graphs with excluded topological subgraphs
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A linear kernel for planar total dominating set. Manuscript available at arxiv.org/abs/1211.0978
, 2012
"... Abstract. A total dominating set of a graph G = (V,E) is a subset D ⊆ V such that every vertex in V is adjacent to some vertex in D. Finding a total dominating set of minimum size is NPhard on planar graphs and W [2]complete on general graphs when parameterized by the solution size. By the metath ..."
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Cited by 2 (2 self)
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Abstract. A total dominating set of a graph G = (V,E) is a subset D ⊆ V such that every vertex in V is adjacent to some vertex in D. Finding a total dominating set of minimum size is NPhard on planar graphs and W [2]complete on general graphs when parameterized by the solution size. By the metatheorem of Bodlaender et al. [FOCS 2009], it follows that there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM 2004], we provide an explicit linear kernel for Total Dominating Set on planar graphs. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominat
Linear kernels on graphs excluding topological minors
"... We show that problems that have finite integer index and satisfy a requirement we call treewidthbounding admit linear kernels on the class ofHtopologicalminor free graphs, for an arbitrary fixed graphH. This builds on earlier results by Bodlaender et al. on graphs of bounded genus [2] and by Fom ..."
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Cited by 2 (0 self)
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We show that problems that have finite integer index and satisfy a requirement we call treewidthbounding admit linear kernels on the class ofHtopologicalminor free graphs, for an arbitrary fixed graphH. This builds on earlier results by Bodlaender et al. on graphs of bounded genus [2] and by Fomin et al. onHminorfree graphs [9]. Our framework encompasses several problems, the prominent ones being Chordal Vertex Deletion, Feedback Vertex Set and Edge
Graph Minors and Parameterized Algorithm Design
"... Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct metaalgorithmic ..."
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Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct metaalgorithmic consequences, we present the algorithmic applications of core theorems such as the gridexclusion theorem, and we give a brief description of the irrelevant vertex technique.
Kernelization and Sparseness: the case of Dominating Set∗
, 2014
"... The search for linear kernels for the Dominating Set problem on classes of graphs of a topological nature has been one of the leading trends in kernelization in recent years. Following the fundamental work of Alber et al. [2] that established a linear kernel for the problem on planar graphs, linear ..."
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The search for linear kernels for the Dominating Set problem on classes of graphs of a topological nature has been one of the leading trends in kernelization in recent years. Following the fundamental work of Alber et al. [2] that established a linear kernel for the problem on planar graphs, linear kernels have been given for boundedgenus graphs [4], apexminorfree graphs [15], Hminorfree graphs [16], and Htopologicalminorfree graphs [17]. These generalizations are based on bidimensionality and powerful decomposition theorems for Hminorfree graphs and Htopologicalminorfree graphs of Robertson and Seymour [28] and of Grohe and Marx [22]. In this work we investigate a new approach to kernelization algorithms for Dominating Set on sparse graph classes. The approach is based on the theory of bounded expansion and nowhere dense graph classes, developed in the recent years by Nešetřil and Ossona de Mendez, among others. More precisely, we prove that Dominating Set admits a linear kernel on any hereditary graph class of bounded expansion and an almost linear kernel on any hereditary nowhere dense graph class. Since the class of Htopologicalminorfree graphs has bounded expansion, our results strongly generalize all the above mentioned works on kernelization of Dominating Set.
(Meta) Kernelization
, 2013
"... In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k, while preserving the answer. In this work we give two metatheorems on kernelzation. ..."
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In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k, while preserving the answer. In this work we give two metatheorems on kernelzation. The first theorem says that all problems expressible in Counting Monadic Second Order Logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus. Our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus. These theorems unify and extend all previously known kernelization results for planar graph problems.