Results 1  10
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20
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.
Parameterized Complexity: Exponential SpeedUp for Planar Graph Problems
 in Electronic Colloquium on Computational Complexity (ECCC
, 2001
"... A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniqu ..."
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Cited by 62 (21 self)
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A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c p k for a large variety of planar graph problems. The key to this type of algorithm is what we call the "Layerwise Separation Property" of a planar graph problem. Problems having this property include planar vertex cover, planar independent set, and planar dominating set.
Algorithms For Vertex Partitioning Problems On Partial kTrees
, 1997
"... In this paper, we consider a large class of vertex partitioning problems and apply to those the theory of algorithm design for problems restricted to partial ktrees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutio ..."
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Cited by 41 (3 self)
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In this paper, we consider a large class of vertex partitioning problems and apply to those the theory of algorithm design for problems restricted to partial ktrees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutions for practical applications.
Fixed parameter algorithms for planar dominating set and related problems
, 2000
"... We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c √ kn), where c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O ( � γ(G)), and that such a tree decomposition ca ..."
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Cited by 35 (10 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 √ kn), where c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O ( � γ(G)), and that such a tree decomposition can be found in O ( � γ(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 √ k in O(c1 n + n2) time, where c1 = 236√34 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. Keywords. NPcomplete problems, fixed parameter tractability, planar graphs, planar dominating set, face cover, outerplanarity, treewidth.
Improved Tree Decomposition Based Algorithms for Dominationlike Problems
 in LATIN’02: Theoretical informatics (Cancun
, 2001
"... We present an improved dynamic programming strategy for dominating set and related problems on graphs that are given together with a tree decomposition of width k. We obtain an O(4 n) algorithm for dominating set, where n is the number of nodes of the tree decomposition. ..."
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Cited by 32 (9 self)
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We present an improved dynamic programming strategy for dominating set and related problems on graphs that are given together with a tree decomposition of width k. We obtain an O(4 n) algorithm for dominating set, where n is the number of nodes of the tree decomposition.
Graph separators: a parameterized view
 Journal of Computer and System Sciences
, 2001
"... Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. ..."
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Cited by 29 (13 self)
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Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p
Complexity of dominationtype problems in graphs
 NORDIC JOURNAL OF COMPUTING
, 1994
"... Many graph parameters are the optimal value of an objective function over selected subsets S of vertices with some constraint on how many selected neighbors vertices in S, and vertices not in S, can have. Classic examples are minimum dominating set and maximum independent set. We give a characteriza ..."
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Cited by 23 (5 self)
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Many graph parameters are the optimal value of an objective function over selected subsets S of vertices with some constraint on how many selected neighbors vertices in S, and vertices not in S, can have. Classic examples are minimum dominating set and maximum independent set. We give a characterization of these graph parameters that unifies their definitions, facilitates their common algorithmic treatment and allows for their uniform complexity classification. This characterization provides the basis for a taxonomy of dominationtype and independencetype problems. We investigate the computational complexity of problems within this taxonomy, identify classes of NPcomplete problems and classes of problems solvable in polynomial time. CR Classification: F.2.2, G.2.2
A Structural View on Parameterizing Problems: Distance from Triviality
 In First International Workshop on Parameterized and Exact Computation, IWPEC 2004, LNCS Proceedings
, 2004
"... Based on a series of known and new examples, we propose the generalized setting of "distance from triviality" measurement as a reasonable and prospective way of determining useful structural problem parameters in analyzing computationally hard problems. The underlying idea is to consid ..."
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Cited by 22 (10 self)
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Based on a series of known and new examples, we propose the generalized setting of "distance from triviality" measurement as a reasonable and prospective way of determining useful structural problem parameters in analyzing computationally hard problems. The underlying idea is to consider tractable special cases of generally hard problems and to introduce parameters that measure the distance from these special cases. In this paper we present several case studies of distance from triviality parameterizations (concerning Clique, Power Dominating Set, Set Cover, and Longest Common Subsequence) that exhibit the versatility of this approach to develop important new views for computational complexity analysis.
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
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