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Approximation Algorithms for Connected Dominating Sets
 Algorithmica
, 1996
"... The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, whe ..."
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Cited by 281 (9 self)
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The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of O(H (\Delta)) are presented, where \Delta is the maximum degree, and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited, or has at least one of its neighbors visited. We study a generalization of the problem when the vertices have weights, and give an algorithm which achieves a performance ratio of 3 ln n. We also consider the ...
Polyhedral studies in Domination Graph Theory (I)
, 2003
"... This paper discusses polyhedral approaches to problems in Domination Graph Thoery. We give various linear integer programming formulations for the weighted and unweighted versions of the minimum dominating set problem. We study the associated polytopes and determine dimension of the polytopes, facet ..."
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This paper discusses polyhedral approaches to problems in Domination Graph Thoery. We give various linear integer programming formulations for the weighted and unweighted versions of the minimum dominating set problem. We study the associated polytopes and determine dimension of the polytopes, facets, valid inequalities et al. Ideas from integer programming such as lift and project are used to derive strong formulations. Polyhedral connections between the dominating set polytope, spanning tree polytope and the matching polytope are established. Integer programming formulations and associated polyhedral results for the weighted versions of
Approximation Algorithms for Connected Dominating Sets
, 1998
"... The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minim ..."
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The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(�) + 2 and H(�) + 2 are presented, where � is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (cn + 1) ln n where cn ln k is the approximation factor for the node weighted Steiner tree problem (currently cn = 1.6103). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c + 1)H(�) + c − 1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644).
Approximation Algorithms for Connected Dominating Sets
, 1996
"... The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, wh ..."
Abstract
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The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of O(H(∆)) are presented, where ∆ is the maximum degree, and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited, or has at least one of its neighbors visited. We study a generalization of the problem when the vertices have weights, and give an algorithm which achieves a performance ratio of 3 ln n. We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide an O(H(∆)) approximation factor. To prove the bound we also develop an optimal approximation algorithm for the unit node weighted Steiner tree problem.
CONNECTED VERTEX COVERS IN DENSE GRAPHS
"... Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parame ..."
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Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parameterized upper bounds on the approximation returned by Savage’s algorithm, and extend a vertex cover algorithm from Karpinski and Zelikovsky to the connected case. The new algorithm approximates the minimum connected vertex cover problem within a factor strictly less than 2 on all dense graphs. All these results are shown to be tight. Finally, we introduce the price of connectivity for the vertex cover problem, defined as the worstcase ratio between the sizes of a minimum connected vertex cover and a minimum vertex cover. We prove that the price of connectivity is bounded by 2/(1 + ε) in graphs with average degree εn, and give a family of neartight examples. Key words: approximation algorithm, vertex cover, connected vertex cover, dense graph. 1.
FINDING INVESTIGATOR TOURS IN TELECOMMUNICATION NETWORKS USING GENETIC ALGORITHMS
"... genetic algorithms, algorithm complexity, graph theory We review and analyze a formal problem of fault detection in point topoint telecommunications networks that are modeled as undirected graphs. Two heuristics, one deterministic and the other an application of genetic algorithm techniques, are tes ..."
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genetic algorithms, algorithm complexity, graph theory We review and analyze a formal problem of fault detection in point topoint telecommunications networks that are modeled as undirected graphs. Two heuristics, one deterministic and the other an application of genetic algorithm techniques, are tested on several sample graphs. The performance of these heuristics is compared and interpreted. The genetic algorithm technique consistently outperforms the deterministic technique on our test data sets.