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67
Fixed Parameter Algorithms for Dominating Set and Related Problems on Planar Graphs
, 2002
"... We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. ..."
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Cited by 112 (22 self)
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We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. The same technique can be used to show that the kface cover problem ( find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c n) time, where c 1 = 3 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of kdominating set, e.g., kindependent dominating set and kweighted dominating set.
Dominating Sets in Planar Graphs: BranchWidth and Exponential Speedup
, 2002
"... Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance. ..."
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Cited by 69 (18 self)
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Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance.
PolynomialTime Data Reduction for DOMINATING SET
 Journal of the ACM
, 2004
"... Dealing with the NPcomplete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a socalled problem kernel of linear size, achiev ..."
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Cited by 64 (8 self)
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Dealing with the NPcomplete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a socalled problem kernel of linear size, achieved by two simple and easy to implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization.
Subexponential parameterized algorithms on graphs of boundedgenus and Hminorfree Graphs
"... ... Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. Inparticular, this general category of graphs includes planar graphs, boundedgenus graphs, singlecrossing ..."
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Cited by 63 (22 self)
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... Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. Inparticular, this general category of graphs includes planar graphs, boundedgenus graphs, singlecrossingminorfree graphs, and anyclass of graphs that is closed under taking minors. Specifically, the running time is 2O(pk)nh, where h is a constant depending onlyon H, which is polynomial for k = O(log² n). We introducea general approach for developing algorithms on Hminorfreegraphs, based on structural results about Hminorfree graphs at the
Parametric duality and kernelization: lower bounds and upper bounds on kernel size
 In Proc. 22nd STACS, volume 3404 of LNCS
, 2005
"... Abstract. Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable ..."
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Cited by 51 (5 self)
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Abstract. Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixedparameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes handinhand with the design of practical algorithms for solving NPhard problems. Wellknown examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by 2k, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by 335k. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless P = NP, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to 67k, improving significantly the 335k previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363–384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.
Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions
 IN PROCEEDINGS OF THE 13TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA 2005
, 2005
"... A divideandconquer strategy based on variations of the LiptonTarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on pla ..."
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Cited by 48 (18 self)
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A divideandconquer strategy based on variations of the LiptonTarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour & Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of noncrossing partitions. Compared to divideandconquer algorithms, the main advantages of our method are a) it is a generic method which allows to attack broad classes of problems; b) the obtained algorithms provide a better worst case analysis. To exemplify our approach we show how to obtain an O(26.903pn) time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time O(29.8594pn). Our approach can be used to design parameterized algorithms as well. For example we introduce the first 2O(pk)nO(1) time algorithm for parameterized Planar kcycle by showing that for a given k we can decide if a planar graph on n vertices has a cycle of length at least k in time O(213.6pkn + n3).
Locally excluding a minor
"... We introduce the concept of locally excluded minors. Graph classes locally excluding a minor are a common generalisation of the concept of excluded minor classes and of graph classes with bounded local treewidth. We show that firstorder modelchecking is fixedparameter tractable on any class of gr ..."
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Cited by 47 (13 self)
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We introduce the concept of locally excluded minors. Graph classes locally excluding a minor are a common generalisation of the concept of excluded minor classes and of graph classes with bounded local treewidth. We show that firstorder modelchecking is fixedparameter tractable on any class of graphs locally excluding a minor. This strictly generalises analogous results by Flum and Grohe on excluded minor classes and Frick and Grohe on classes with bounded local treewidth. As an important consequence of the proof we obtain fixedparameter algorithms for problems such as dominating or independent set on graph classes excluding a minor, where now the parameter is the size of the dominating set and the excluded minor. We also study graph classes with excluded minors, where the minor may grow slowly with the size of the graphs and show that again, firstorder modelchecking is fixedparameter tractable on any such class of graphs.
The bidimensionality Theory and Its Algorithmic Applications
 Computer Journal
, 2005
"... This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixedparameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, boundedgenus graphs and gra ..."
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Cited by 47 (3 self)
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This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixedparameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, boundedgenus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the Graph Minor Theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. Here, we summarize the known combinatorial and algorithmic results of bidimensionality theory with the highlevel ideas involved in their proof; we describe the previous work on which the theory is based and/or extends; and we mention several remaining open problems. 1.
On Efficient FixedParameter Algorithms for Weighted Vertex Cover
"... this paper was presented at the 11th Annual International Symposium on Algorithms And Computation (ISAAC'00), SpringerVerlag, LNCS 1969, pages 180191, held in Taipei, Taiwan, December 2000. This conference version, however, contains a faulty application of the main result to the case of mini ..."
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Cited by 45 (14 self)
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this paper was presented at the 11th Annual International Symposium on Algorithms And Computation (ISAAC'00), SpringerVerlag, LNCS 1969, pages 180191, held in Taipei, Taiwan, December 2000. This conference version, however, contains a faulty application of the main result to the case of minimum weight vertex covers with a bound on the number of vertices
Bidimensionality: New Connections between FPT Algorithms and PTASs
, 2005
"... We demonstrate a new connection between fixedparametertractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled "bidimensional " problems to show that essentially ..."
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Cited by 43 (7 self)
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We demonstrate a new connection between fixedparametertractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled &quot;bidimensional &quot; problems to show that essentially all such problems have both subexponential fixedparameter algorithms and PTASs. Bidimensional problems include e.g. feedbackvertex set, vertex cover, minimum maximal matching, face cover, a series of vertexremoval problems, dominating set,edge dominating set,