On approximate optimal dual power assignment for biconnectivity and edge-biconnectivity

Topology control is one of the major approaches to achieve the energy efficiency as well as fault tolerance in wireless networks. In this paper, we study the dual power assignment problem for 2-edge connectivity and 2vertex connectivity in the symmetric graphical model. The problem is arisen from the following practical origin. In a wireless ad hoc network where each node can switch its transmission power between high-level and low-level, how can we establish a fault-tolerantly connected network topology in the most energy-efficient way? Specifically, the objective is to minimize the number of nodes assigned with high power and yet achieve 2-edge connectivity or 2-vertex connectivity. Note that to achieve a minimum number of high-power nodes is harder than an optimization problem in the same model whose objective is to minimize the total power cost. We first address these two optimization problems (2-edge connectivity and 2-vertex connectivity version) under the general graph model. Due to the NP-hardness, we propose an approximation algorithm, called prioritized edge selection algorithm, which achieves a 4-ratio approximation for 2-edge connectivity. After that, we modify the algorithm to solve the problem for 2vertex connectivity and also achieve the same approximation ratio. We also show that the 4-ratio is tight for our algorithms in both cases.