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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.
Deciding firstorder properties of nowhere dense graphs
 In Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC
, 2014
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Enumerating with constant delay the answers to a query
"... We survey recent results about enumerating with constant delaytheanswerstoaqueryoveradatabase. More precisely, we focus on the case when enumeration can be achieved with a preprocessing running in time linear in the size of the database, followed by an enumeration process outputting the answers one ..."
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We survey recent results about enumerating with constant delaytheanswerstoaqueryoveradatabase. More precisely, we focus on the case when enumeration can be achieved with a preprocessing running in time linear in the size of the database, followed by an enumeration process outputting the answers one by one with constant time between any consecutive outputs. We survey classes of databases and classes of queries for which this is possible. We also mention related problems such as computing the number of answers or sampling the set of answers.
Characterisations of Nowhere Dense Graphs
, 2013
"... Nowhere dense classes of graphs were introduced by Nešetřil and Ossona de Mendez as a model for “sparsity” in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) ..."
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Nowhere dense classes of graphs were introduced by Nešetřil and Ossona de Mendez as a model for “sparsity” in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) model theory. Therefore, the concept of nowhere density seems to capture a natural property of graph classes generalising for example classes of graphs which exclude a fixed minor, have bounded degree or bounded local treewidth. In this paper we give a selfcontained introduction to the concept of nowhere dense classes of graphs focussing on the various ways in which they can be characterised. We also briefly sketch algorithmic applications these characterisations have found in the literature.
A glimpse on constant delay enumeration
"... We survey some of the recent results about enumerating the answers to queries over a database. We focus on the case where the enumeration is performed with a constant delay between any two consecutive solutions, after a linear time preprocessing. This cannot be always achieved. It requires restricti ..."
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We survey some of the recent results about enumerating the answers to queries over a database. We focus on the case where the enumeration is performed with a constant delay between any two consecutive solutions, after a linear time preprocessing. This cannot be always achieved. It requires restricting either the class of queries or the class of databases. We describe here several scenarios when this is possible.
Hyperbolicity, degeneracy, and expansion of random intersection graphs
, 2014
"... We determine several key structural features of random intersection graphs, a natural model for many real world networks where connections are given by shared attributes. Notably, this model is mathematically tractable yet flexible enough to generate random graphs with tunable clustering. Specifical ..."
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We determine several key structural features of random intersection graphs, a natural model for many real world networks where connections are given by shared attributes. Notably, this model is mathematically tractable yet flexible enough to generate random graphs with tunable clustering. Specifically, we prove that in the homogeneous case, the model is logarithmically hyperbolic. Further, we fully characterize the degeneracy and the expansion of homogeneous random intersection graphs. Some of these results apply to simple inhomogeneous random intersection graphs. Finally, we comment on the algorithmic implications of these results. 1
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.