Greedy approximation for the source location problem with vertex-connectivity requirements in undirected graphs |
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Authors: | Toshimasa Ishii |
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Institution: | aDepartment of Information and Management Science, Otaru University of Commerce, Otaru-city, Hokkaido 047-8501, Japan |
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Abstract: | Let be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex has a demand , and a cost , where and denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G asks to find a set S of vertices minimizing such that there are at least pairwise vertex-disjoint paths from S to v for each vertex . It is known that the problem is not approximable within a ratio of , unless NP has an -time deterministic algorithm. Also, it is known that even if every vertex has a uniform cost and holds, then the problem is NP-hard, where .In this paper, we consider the problem in the case where every vertex has uniform cost. We propose a simple greedy algorithm for providing a -approximate solution to the problem in time, while we also show that there exists an instance for which it provides no better than a -approximate solution. Especially, in the case of , we give a tight analysis to show that it achieves an approximation ratio of 3. We also show the APX-hardness of the problem even restricted to . |
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Keywords: | Graph algorithm Greedy algorithm Undirected graph Location problem Vertex-connectivity |
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