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Abstract:

Depending on whether bidirectional links or unidirectional links are used for communications, the network topology under a given range assignment is either an undirected graph referred to as the symmetric topology, or a directed graph referred to as the asymmetric topology. The Min-Power Symmetric (resp., Asymmetric) k-Node Connectivity problem seeks a range assignment of minimum total power subject to the constraint the induced symmetric (resp. asymmetric) topology is k-connected. Similarly, the Min-Power Symmetric (resp., Asymmetric) k-Edge Connectivity problem seeks a range assignment of minimum total power subject to the constraint the induced symmetric (resp., asymmetric) topology is k-edge connected. The Min-Power Symmetric Biconnectivity problem and the Min-Power Symmetric EdgeBiconnectivity problem has been studied by Lloyd et. al [22]. They show that range assignment based the approximation algorithm of Khuller and Raghavachari [18], which we refer to as Algorithm KR, has an approximation ratio of at most 2(2 2=n)(2 + 1=n) for Min-Power Symmetric Biconnectivity, and range assignment based on the approximation algorithm of Khuller and Vishkin [19], which we refer to as Algorithm KV, has an approximation ratio of at most 8(1 1=n) for Min-Power Symmetric Edge-Biconnectivity. In this paper, we rst establish the NP-hardness of Min-Power Symmetric (Edge-)Biconnectivity. Then we show that Algorithm KR has an approximation ratio of at most 4 for both Min-Power Symmetric Biconnectivity and Min-Power Asymmetric Biconnectivity, and Algorithm KV has an approximation ratio of at most 2k for both Min-Power Symmetric k-Edge Connectivity and MinPower Asymmetric k-Edge Connectivity. We also propose a new simple constant-approximation algorithm for both Min-Power Symmetric Biconnectivity and Min-Power Asymmetric Biconnectivity. This new algorithm is best suited for distributed implementation. 1

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