Results 1  10
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83
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 68 (14 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.
Subexponential Parameterized Algorithms on Graphs of Bounded Genus and HMinorFree Graphs
, 2003
"... We introduce a new framework for designing fixedparameter algorithms with subexponential running time2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum m ..."
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Cited by 41 (13 self)
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We introduce a new framework for designing fixedparameter algorithms with subexponential running time2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, cliquetransversal set, and many others restricted to bounded genus graphs. Furthermore, it is fairly straightforward to prove that a problem is bidimensional. In particular, our framework includes as special cases all previously known problems to have such subexponential algorithms. Previously, these algorithms applied to planar graphs, singlecrossingminorfree graphs, and/or map graphs; we extend these results to apply to boundedgenus graphs as well. In a parallel development of combinatorial results, we establish an upper bound on the treewidth (or branchwidth) of a boundedgenus graph that excludes some planar graph H as a minor. This bound depends linearly on the size (H) of the excluded graph H and the genus g(G) of the graph G, and applies and extends the graphminors work of Robertson and Seymour. Building on these results...
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 39 (17 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).
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 39 (9 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.
Tight lower bounds for certain parameterized NPhard problems
 Information and Computation
, 2004
"... Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of wellknown NPhard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on deptht circuits cannot be solve ..."
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Cited by 37 (5 self)
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Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of wellknown NPhard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on deptht circuits cannot be solved in time n o(k) m O(1) , where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t − 1)st level W [t − 1] of the Whierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NPhard problems, including weighted sat, hitting set, set cover, and feature set, cannot be solved in time n o(k) m O(1) , where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W [1] of the Whierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted qsat (for any fixed q ≥ 2), clique, independent set, and dominating set, cannot be solved in time n o(k) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n k m O(1) or O(n k). 1
Bidimensionality: New Connections between FPT Algorithms and PTASs
"... We demonstrate a new connection between fixedparameter tractability 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 all such problems ha ..."
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Cited by 36 (5 self)
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We demonstrate a new connection between fixedparameter tractability 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 all such problems have both subexponential fixedparameter algorithms and PTASs. Bidimensional problems include e.g. feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertexremoval problems, dominating set, edge dominating set, rdominating set, diameter, connected dominating set, connected edge dominating set, and connected rdominating set. We obtain PTASs for all of these problems in planar graphs and certain generalizations; of particular interest are our results for the two wellknown problems of connected dominating set and general feedback vertex set for planar graphs and their generalizations, for which PTASs were not known to exist. Our techniques generalize and in some sense unify the two main previous approaches for designing PTASs in planar graphs, namely, the LiptonTarjan separator approach [FOCS’77] and the Baker layerwise decomposition approach [FOCS’83]. In particular, we replace the notion of separators with a more powerful tool from the bidimensionality theory, enabling the first approach to apply to a much broader class of minimization problems than previously possible; and through the use of a structural backbone and thickening of layers we demonstrate how the second approach can be applied to problems with a “nonlocal” structure.
Parameterized Complexity for the Skeptic
 In Proc. 18th IEEE Annual Conference on Computational Complexity
, 2003
"... The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity. ..."
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Cited by 36 (1 self)
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The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity.
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 35 (4 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.
Fixedparameter algorithms for the (k, r)center in planar graphs and map graphs
 ACM TRANSACTIONS ON ALGORITHMS
, 2003
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