Results 1  10
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28
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 105 (23 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 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.
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.
Efficient Data Reduction for Dominating Set: A Linear Problem Kernel for the Planar Case (Extended Abstract)
 Lecture Notes in Computer Science (LNCS
, 2002
"... Dealing with the NPcomplete Dominating Set problem on undirected 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 on planar graphs has a socalled problem kernel of linear size, achieved ..."
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Cited by 22 (8 self)
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Dealing with the NPcomplete Dominating Set problem on undirected 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 on planar graphs has a socalled problem kernel of linear size, achieved by two simple and easy to implement reduction rules. This answers an open question from previous work on the parameterized complexity of Dominating Set on planar graphs.
FPT is Ptime extremal structure I
 Algorithms and Complexity in Durham 2005, Proceedings of the first ACiD Workshop, volume 4 of Texts in Algorithmics
, 2005
"... We describe a broad program of research in parameterized complexity, and hows this plays out for the MAX LEAF SPANNING TREE problem. ..."
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Cited by 20 (1 self)
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We describe a broad program of research in parameterized complexity, and hows this plays out for the MAX LEAF SPANNING TREE problem.
Branch and Tree Decomposition Techniques for Discrete Optimization
, 2005
"... This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connecti ..."
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Cited by 16 (3 self)
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This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NPhard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixedparameter tractable algorithms and have been shown to be effective in a practical setting for NPhard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.
Improved algorithms and complexity results for power domination
 in graphs, Lecture Notes Comp. Sci. 3623
, 2005
"... Abstract. The Power Dominating Set problem is a variant of the classical domination problem in graphs: Given an undirected graph G = (V, E), find a minimum P ⊆ V such that all vertices in V are “observed” by vertices in P. Herein, a vertex observes itself and all its neighbors, and if an observed ve ..."
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Cited by 13 (2 self)
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Abstract. The Power Dominating Set problem is a variant of the classical domination problem in graphs: Given an undirected graph G = (V, E), find a minimum P ⊆ V such that all vertices in V are “observed” by vertices in P. Herein, a vertex observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. We show that Power Dominating Set can be solved by “boundedtreewidth dynamic programs.” Moreover, we simplify and extend several NPcompleteness results, particularly showing that Power Dominating Set remains NPcomplete for planar graphs, for circle graphs, and for split graphs. Specifically, our improved reductions imply that Power Dominating Set parameterized by P  is W[2]hard and cannot be better approximated than Dominating Set. 1
Experimental evaluation of a tree decomposition based algorithm for vertex cover on planar graphs
 Disc. Appl. Math
, 2004
"... Many NPcomplete problems on planar graphs are “fixedparameter tractable:” Recent theoretical work provided tree decomposition based fixedparameter algorithms exactly solving various parameterized problems on planar graphs, among others Vertex Cover, in time O(c √ k n). Here, c is some constant de ..."
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Cited by 13 (6 self)
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Many NPcomplete problems on planar graphs are “fixedparameter tractable:” Recent theoretical work provided tree decomposition based fixedparameter algorithms exactly solving various parameterized problems on planar graphs, among others Vertex Cover, in time O(c √ k n). Here, c is some constant depending on the graph problem to be solved, n is the number of graph vertices, and k is the problem parameter (for Vertex Cover this is the size of the vertex cover). In this paper, we present an experimental study for such tree decomposition based algorithms focusing on Vertex Cover. We demonstrate that the tree decomposition based approach provides a valuable way of exactly solving Vertex Cover on planar graphs. Doing so, we also demonstrate the impressive power of the socalled Nemhauser/Trotter theorem which provides a Vertex Coverspecific, extremely useful data reduction through polynomial time preprocessing. Altogether, this underpins the practical importance of the underlying theory. 1
Either/or: Using vertex cover structure in designing FPTalgorithms—the case of kinternal spanning tree
 In Proceedings of WADS 2003, Workshop on Algorithms and Data Structures, volume 2748 of LNCS
, 2003
"... Abstract. To determine if a graph has a spanning tree with few leaves is NPhard as HAMILTONIAN PATH is a special case. In this paper we study the parametric dual of this problem, kINTERNAL SPANNING TREE (Does G have a spanning tree with at least k internal vertices?). We give an algorithm running ..."
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Cited by 11 (1 self)
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Abstract. To determine if a graph has a spanning tree with few leaves is NPhard as HAMILTONIAN PATH is a special case. In this paper we study the parametric dual of this problem, kINTERNAL SPANNING TREE (Does G have a spanning tree with at least k internal vertices?). We give an algorithm running in time O(2 4k log k · k 7/2 + k 2 · n 2). We also give a 2approximation algorithm for the problem. However, the main contribution of this paper is that we show the following remarkable structural bindings between kINTERNAL SPANNING TREE and kVERTEX COVER: • NO for kVERTEX COVER implies YES for kINTERNAL SPANNING TREE. • YES for kVERTEX COVER implies NO for (2k + 1)INTERNAL SPANNING TREE. We give a polynomialtime algorithm that produces either a vertex cover of size k or a spanning tree with at least k internal vertices. We show how to use this inherent vertex cover structure to design algorithms for FPT problems, here illustrated mainly by kINTERNAL SPANNING TREE. We also briefly discuss the application of this vertex cover methodology to the parametric dual of the DOMINATING SET problem. This design technique seems to apply to many other FPT problems.