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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.
Improved Approximation Bounds for Edge Dominating Set in Dense Graphs
"... Abstract. We analyze the simple greedy algorithm that iteratively removes the endpoints of a maximumdegree edge in a graph, where the degree of an edge is the sum of the degrees of its endpoints. This algorithm provides a 2approximation to the minimum edge dominating set and minimum maximal matchi ..."
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Abstract. We analyze the simple greedy algorithm that iteratively removes the endpoints of a maximumdegree edge in a graph, where the degree of an edge is the sum of the degrees of its endpoints. This algorithm provides a 2approximation to the minimum edge dominating set and minimum maximal matching problems. We refine its analysis and give an expression of the approximation ratio that is strictly less than 2 in the cases where the input graph has n vertices and at edges, for ɛ> 1/2. This ratio is shown to be asymptotically tight for least ɛ ` n 2 ɛ> 1/2.
Integer Programming Formulations for the Minimum Weighted Maximal Matching Problem
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"... Given an undirected graph, the problem of finding a maximal matching that has minimum total weight is NPhard. This problem has been studied extensively from a graph theoretical point of view. Most of the existing literature considers the problem in some restricted classes of graphs and give polyn ..."
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Cited by 2 (2 self)
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Given an undirected graph, the problem of finding a maximal matching that has minimum total weight is NPhard. This problem has been studied extensively from a graph theoretical point of view. Most of the existing literature considers the problem in some restricted classes of graphs and give polynomial time exact or approximation algorithms. On the contrary, we consider the problem on general graphs and approach it from an optimization point of view. In this paper, we develop integer programming formulations for the minimum weighted maximal matching problem and analyze their efficacy on randomly generated graphs. We also compare solutions found by a greedy approximation algorithm, which is based on the literature, against optimal solutions. Our results show that our integer programming formulations are able to solve medium size instances to optimality and suggest further research for improvement.