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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
Kernelization: New Upper and Lower Bound Techniques
 In Proc. of the 4th International Workshop on Parameterized and Exact Computation (IWPEC), volume 5917 of LNCS
, 2009
"... Abstract. In this survey, we look at kernelization: algorithms that transform in polynomial time an input to a problem to an equivalent input, whose size is bounded by a function of a parameter. Several results of recent research on kernelization are mentioned. This survey looks at some recent resu ..."
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Cited by 54 (0 self)
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Abstract. In this survey, we look at kernelization: algorithms that transform in polynomial time an input to a problem to an equivalent input, whose size is bounded by a function of a parameter. Several results of recent research on kernelization are mentioned. This survey looks at some recent results where a general technique shows the existence of kernelization algorithms for large classes of problems, in particular for planar graphs and generalizations of planar graphs, and recent lower bound techniques that give evidence that certain types of kernelization algorithms do not exist.
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.
 In Proc. of the 40th International Colloquium on Automata, Languages and Programming (ICALP),
, 2013
"... Abstract 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 ex ..."
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Cited by 15 (4 self)
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Abstract 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 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 Our second application concerns the PlanarFDeletion problem. Let F be a fixed finite family of graphs containing at least one planar graph. Given an nvertex graph G and a nonnegative integer k, PlanarFDeletion asks whether G has a set X ⊆ V (G) such that X k and G − X is Hminorfree for every H ∈ F. This problem encompasses a number of wellstudied parameterized problems such as Vertex Cover, Feedback Vertex Set, and Treewidtht Vertex Deletion. Very recently, an algorithm for PlanarFDeletion with running time 2 O(k) · n log 2 n (such an algorithm is called singleexponential) has been presented in
The parameterized complexity of the induced matching problem in planar graphs
 In Proceedings of the 2007 International Frontiers of Algorithmics Workshop, Lecture Notes in Comput. Sci
, 2007
"... Given a graph G and an integer k ≥ 0, the NPcomplete Induced Matching problem asks whether there exists an edge subset M of size at least k such that M is a matching and no two edges of M are joined by an edge of G. The complexity of this problem on general graphs as well as on many restricted grap ..."
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Cited by 15 (1 self)
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Given a graph G and an integer k ≥ 0, the NPcomplete Induced Matching problem asks whether there exists an edge subset M of size at least k such that M is a matching and no two edges of M are joined by an edge of G. The complexity of this problem on general graphs as well as on many restricted graph classes has been studied intensively. However, other than the fact that the problem is W[1]hard on general graphs little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we provide firsttime fixedparameter tractability results for planar graphs, boundeddegree graphs, graphs with girth at least six, bipartite graphs, line graphs, and graphs of bounded treewidth. In particular, we give a linearsize problem kernel for planar graphs.
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.
A moderately exponential time algorithm for full degree spanning tree
 in the proceedings of TAMC 2008, LNCS 4978
, 2008
"... We present a moderately exponential time exact algorithm for the wellstudied Full Degree Spanning Tree problem, an NPhard variant of the Spanning Tree problem. Given a graph G, the objective is to find a spanning tree T of G which maximizes the number of vertices that have the same degree in T as i ..."
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Cited by 4 (2 self)
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We present a moderately exponential time exact algorithm for the wellstudied Full Degree Spanning Tree problem, an NPhard variant of the Spanning Tree problem. Given a graph G, the objective is to find a spanning tree T of G which maximizes the number of vertices that have the same degree in T as in G. The problem is motivated by its application in fluid networks and is basically a graphtheoretic abstraction of the problem of placing flow meters in fluid networks. We give an exact algorithm for Full Degree Spanning Tree running in time O(1.9465 n). This adds Full Degree Spanning Tree to a very small list of “nonlocal problems”, like Feedback Vertex Set and Connected Dominating Set, for which nontrivial (non brute force enumeration) exact algorithms are known. 1
On the Directed DegreePreserving Spanning Tree Problem
"... Abstract. In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree with at most k vertices of reduced outdegree. This problem is a directed analog of the wellstudied Mi ..."
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Abstract. In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree with at most k vertices of reduced outdegree. This problem is a directed analog of the wellstudied MinimumVertex Feedback Edge Set problem. We show that this problem is fixedparameter tractable and admits a problem kernel with at most 8k vertices on strongly connected digraphs and O(k 2) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with runtime O ∗ (5.942 k). This adds the Reduced Degree Spanning Tree problem to the small list of directed graph problems for which fixedparameter tractable algorithms are known. Finally, we consider the dual of Reduced Degree Spanning Tree, that is, given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree of D with at least k vertices of full outdegree. We show that this problem is W[1]hard on two important digraph classes: directed acyclic graphs and strongly connected digraphs. 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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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
On the Directed Full Degree Spanning Tree Problem
"... Abstract. We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree T of D such that at least k vertices in T have the same outdegree as in D. We show that t ..."
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Abstract. We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree T of D such that at least k vertices in T have the same outdegree as in D. We show that this problem is W[1]hard even on the class of directed acyclic graphs. In the dual version, called Reduced Degree Spanning Tree, one is required to construct a spanning outtree T such that at most k vertices in T have outdegrees that are different from that in D. We show that this problem is fixedparameter tractable and that it admits a problem kernel with at most 8k vertices on strongly connected digraphs and O(k 2 ) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with running time O(5.942 k · n O(1) ), where n is the number of vertices in the input digraph.