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13
Bidimensionality and Kernels
, 2010
"... Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi ..."
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Cited by 58 (23 self)
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Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this paper, we establish a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In parameterized complexity, each problem instance comes with a parameter k and the parameterized problem is said to admit a linear kernel if there is a polynomial time algorithm, called
Linear problem kernels for NPhard problems on planar graphs
 In Proc. 34th ICALP, volume 4596 of LNCS
, 2007
"... Abstract. We develop a generic framework for deriving linearsize problem kernels for NPhard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for Connected Vertex Cover, Minimum Edge Dominating Set, Maximum ..."
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Cited by 32 (5 self)
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Abstract. We develop a generic framework for deriving linearsize problem kernels for NPhard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for Connected Vertex Cover, Minimum Edge Dominating Set, Maximum Triangle Packing, and Efficient Dominating Set on planar graphs. On the route to these results, we present effective, problemspecific data reduction rules that are useful in any approach attacking the computational intractability of these problems. 1
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 15 (4 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.
ON THE INDUCED MATCHING PROBLEM
, 2008
"... We study extremal questions on induced matchings in several natural graph classes. We argue that these questions should be asked for twinless graphs, that is graphs not containing two vertices with the same neighborhood. We show that planar twinless graphs always contain an induced matching of siz ..."
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Cited by 10 (1 self)
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We study extremal questions on induced matchings in several natural graph classes. We argue that these questions should be asked for twinless graphs, that is graphs not containing two vertices with the same neighborhood. We show that planar twinless graphs always contain an induced matching of size at least n/40 while there are planar twinless graphs that do not contain an induced matching of size (n + 10)/27. We derive similar results for outerplanar graphs and graphs of bounded genus. These extremal results can be applied to the area of parameterized computation. For example, we show that the induced matching problem on planar graphs has a kernel of size at most 40k that is computable in linear time; this significantly improves the results of Moser and Sikdar (2007). We also show that we can decide in time O(91 k + n) whether a planar graph contains an induced matching of size at least k.
Linear Kernel for Planar Connected Dominating Set
"... We provide polynomial time data reduction rules for Connected Dominating Set in planar graphs and analyze these to obtain a linear kernel for the planar Connected Dominating Set problem. To obtain the desired kernel we introduce a method that we call reduce or refine. Our kernelization algorithm a ..."
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Cited by 7 (0 self)
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We provide polynomial time data reduction rules for Connected Dominating Set in planar graphs and analyze these to obtain a linear kernel for the planar Connected Dominating Set problem. To obtain the desired kernel we introduce a method that we call reduce or refine. Our kernelization algorithm analyzes the input graph and either finds an appropriate reduction rule that can be applied, or zooms in on a region of the graph which is more amenable to reduction. We find this method of independent interest and believe that it will be useful to obtain linear kernels for other problems on planar graphs.
Towards optimal kernel for connected vertex cover in planar graphs
 CoRR
, 2011
"... Abstract. We study the parameterized complexity of the connected version of the vertex cover problem, where the solution set has to induce a connected subgraph. Although this problem does not admit a polynomial kernel for general graphs (unless NP ⊆ coNP/poly), for planar graphs Guo and Niedermeier ..."
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Cited by 4 (2 self)
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Abstract. We study the parameterized complexity of the connected version of the vertex cover problem, where the solution set has to induce a connected subgraph. Although this problem does not admit a polynomial kernel for general graphs (unless NP ⊆ coNP/poly), for planar graphs Guo and Niedermeier [ICALP’08] showed a kernel with at most 14k vertices, subsequently improved by Wang et al. [MFCS’11] to 4k. The constant 4 here is so small that a natural question arises: could it be already an optimal value for this problem? In this paper we answer this quesion in negative: we show a 11 3 kvertex kernel for Connected Vertex Cover in planar graphs. We believe that this result will motivate further study in search for an optimal kernel. 1
Breaking the 2^nBarrier for Irredundance: Two Lines of Attack
, 2011
"... The lower and the upper irredundance numbers of a graph G, denoted ir(G) and IR(G), respectively, are conceptually linked to the domination and independence numbers and have numerous relations to other graph parameters. It has been an open question whether determining these numbers for a graph G on ..."
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Cited by 3 (2 self)
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The lower and the upper irredundance numbers of a graph G, denoted ir(G) and IR(G), respectively, are conceptually linked to the domination and independence numbers and have numerous relations to other graph parameters. It has been an open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time faster than the trivial Θ(2n · poly(n)) enumeration, also called the 2nbarrier. The main contributions of this article are exact exponentialtime algorithms breaking the 2 nbarrier for irredundance. We establish algorithms with running times of O ∗ (1.99914 n) for computing ir(G) and O ∗ (1.9369 n) for computing IR(G). Both algorithms use polynomial space. The first algorithm uses a parameterized approach to obtain (faster) exact algorithms. The second one is based, in addition, on a reduction to the Maximum Induced Matching problem providing a branchandreduce algorithm to solve it.
IMPROVED INDUCED MATCHINGS IN SPARSE GRAPHS
, 2009
"... An induced matching in graph G is a matching which is an induced subgraph of G. Clearly, among two vertices with the same neighborhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other v ..."
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Cited by 2 (1 self)
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An induced matching in graph G is a matching which is an induced subgraph of G. Clearly, among two vertices with the same neighborhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj, Pelsmajer, Schaefer and Xia [10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least n 40 and that there are twinless planar graphs that do not contain an induced matching of size greater than n + O(1). We improve both these bounds to n +O(1), which is tight up to an additive constant. This implies 28 that the problem of deciding an whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is FPT for graphs of bounded arboricity. Kanj et al. presented also an algorithm which decides in O(2159√k +n)time whether an nvertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we show also a more efficient, O(226√k +n)time algorithm based on the branchwidth decomposition. 27 1
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