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12
Thilikos. Linear kernels for (Connected) Dominating Set on Hminorfree graphs
 In Proceedings of the 22nd Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2012
"... Abstract In the DOMINATING SET problem we are given an nvertex graph G with a positive integer k and we ask whether there exists a vertex subset D of size at most k such that every vertex of G is either in D or is adjacent to some vertex of D. In the connected variant, CONNECTED DOMINATING SET, we ..."
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Abstract In the DOMINATING SET problem we are given an nvertex graph G with a positive integer k and we ask whether there exists a vertex subset D of size at most k such that every vertex of G is either in D or is adjacent to some vertex of D. In the connected variant, CONNECTED DOMINATING SET, we also demand the subgraph induced by D to be connected. Both variants are basic graph problems, known to be NPcomplete, and many algorithmic approaches have been tried on them. In this paper we study both problems on graphs excluding a fixed graph H as a minor from the kernelization point of view. Our main results are polynomial time algorithms that, for a given Hminor free graph G and positive integer k, output an Hminor free graph G 0 on O(k) vertices such that G has a (connected) dominating set of size k if and only if G 0 has. The polynomial time algorithm that obtains such equivalent instance is known as kernelization algorithm and its output is called a problem kernel. If the size of the output can be bounded by a polynomial (linear) function of k, then it is called polynomial (linear) kernel. Prior to our work, the only polynomial kernel for DOMINATING SET on graphs excluding a fixed graph H as a minor was due to Alon and Gutner [ECCC 2008, IWPEC 2009 and to Philip, Raman, and Sikdar [ESA 2009] but the size of their kernel is k c(H) , where c(H) is a constant depending on the size of H. Alon and Gutner asked explicitly, whether one can obtain a linear kernel for DOMINATING SET on Hminor free graphs. We answer this question in affirmative. For CONNECTED DOMINATING SET no polynomial kernel on Hminor free graphs was known prior to our work. As a byproduct of our results we also obtain the first subexponentail time algorithm for CONNECTED DOMINATING SET running in time 2
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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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
Explicit linear kernels via dynamic programming
 IN STACS, VOLUME 25 OF LIPICS
, 2014
"... Several algorithmic metatheorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding ..."
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Cited by 4 (2 self)
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Several algorithmic metatheorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions but, manly due to their generality, it is not known how to derive from them constructive kernels with explicit constants. In this paper we make a step toward a fully constructive metakernelization theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in CMSO logic, uses dynamic programming, which allows us to find an explicit upper bound on the size of the derived kernels. We demonstrate the usefulness of our techniques by providing the first explicit linear kernels for rDominating Set and rScattered Set on apexminorfree graphs, and for PlanarFDeletion and PlanarFPacking on graphs excluding a fixed (topological) minor in the case where all the graphs in F are connected.
PlanarF Deletion: Approximation and Optimal FPT Algorithms
"... Let F be a finite set of graphs. In the FDeletion problem, we are given an nvertex, medge graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by ..."
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Let F be a finite set of graphs. In the FDeletion problem, we are given an nvertex, medge graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Treewidth ηDeletion. In this paper we obtain a number of generic algorithmic results about FDeletion, when F contains at least one planar graph. The highlights of our work are • A randomized O(nm) time constant factor approximation algorithm for the optimization version of FDeletion. • A randomized O(2O(k)n) parameterized algorithm for FDeletion when F is connected. Here a family F is called connected if every graph in F is connected. The algorithm can be made deterministic at the cost of making the polynomial factor in the running time n log2 n rather than linear. These algorithms unify, generalize, and improve over a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results – constant factor approximation and FPT algorithms – are stringed together by a common theme of polynomial time preprocessing. 1
Bidimensionality of Geometric Intersection Graphs
, 2014
"... Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is ..."
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Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is nonempty. For such graph classes and under natural restrictions on their maximum degree or subgraph exclusion, we prove that the relation between their treewidth and the maximum size of a grid minor is linear. These combinatorial results vastly extend the applicability of all the metaalgorithmic results of the bidimensionality theory to geometrically defined graph classes.
Subexponential Parameterized Odd Cycle Transversal on Planar Graphs
"... In the Odd Cycle Transversal (OCT) problem we are given a graph G on n vertices and an integer k, and the objective is to determine whether there exists a vertex set O in G of size at most k such that G \ O is bipartite. Reed, Smith, and Vetta [Oper. Res. Lett., 2004] gave an algorithm for OCT with ..."
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In the Odd Cycle Transversal (OCT) problem we are given a graph G on n vertices and an integer k, and the objective is to determine whether there exists a vertex set O in G of size at most k such that G \ O is bipartite. Reed, Smith, and Vetta [Oper. Res. Lett., 2004] gave an algorithm for OCT with running time 3 k n O(1). Assuming the exponential time hypothesis of Impagliazzo, Paturi and Zane, the running time cannot be improved to 2 o(k) n O(1). We show that OCT admits a randomized algorithm running in O(n O(1) + 2 O( √ k log k) n) time when the input graph is planar. As a byproduct we also obtain a linear time algorithm for OCT on planar graphs with running time O(2 O(k log k) n) time. This improves over an algorithm of Fiorini et
WellQuasiOrders in Subclasses of Bounded Treewidth Graphs and their Algorithmic Applications
"... We show that three subclasses of bounded treewidth graphs are wellquasiordered by refinements of the minor order. Specifically, we prove that graphs with bounded vertexcovers are well quasi ordered by the induced subgraph order, graphs with bounded feedbackvertexset are well quasi ordered by t ..."
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We show that three subclasses of bounded treewidth graphs are wellquasiordered by refinements of the minor order. Specifically, we prove that graphs with bounded vertexcovers are well quasi ordered by the induced subgraph order, graphs with bounded feedbackvertexset are well quasi ordered by the topologicalminor order, and graphs with bounded circumference are well quasi ordered by the inducedminor order. Our results give algorithms for recognizing any graph family in these classes which is closed under the corresponding minor order refinement.
The Directed Grid Theorem
, 2014
"... The grid theorem, originally proved by Robertson and Seymour in Graph Minors V in 1986, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance in bidimensionality theory, and it is the basis for several ..."
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The grid theorem, originally proved by Robertson and Seymour in Graph Minors V in 1986, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance in bidimensionality theory, and it is the basis for several other structure theorems developed in the graph minors project. In the mid90s, Reed and Johnson, Robertson, Seymour and Thomas (see [26, 18]), independently, conjectured an analogous theorem for directed graphs, i.e. the existence of a function f: N → N such that every digraph of directed treewidth at least f(k) contains a directed grid of order k. In an unpublished manuscript from 2001, Johnson, Robertson, Seymour and Thomas give a proof of this conjecture for planar digraphs. But for over a decade, this was the most general case proved for the Reed, Johnson, Robertson, Seymour and Thomas conjecture. Only very recently, this result has been extended to all classes of digraphs excluding a fixed undirected graph as a minor (see [22]). In this paper, nearly two decades after the conjecture was made, we are finally able to confirm the Reed, Johnson, Robertson, Seymour and Thomas conjecture in full generality and to prove the directed grid theorem. As consequence of our results we are able to improve results in Reed et al. in 1996 [28] (see also [25]) on disjoint cycles of length at least l and in [20] on quarterintegral disjoint paths. We expect many more algorithmic results to follow from the grid theorem.