Minimizing the number of max-power users in ad-hoc wireless networks with minimum node degree requirements

We consider the problem of assigning one of two possible transmission ranges to each node of a wireless ad-hoc network, aiming for a predefined degree of redundancy and reliability by requiring that each node has at least Δ∈N neighbors in the resulting network. To conserve the networks limited energy, this goal should be achieved with as few as possible nodes using the higher of the two transmission ranges. We show that this so-called Two Power Level Vertex Degree problem in NP-complete, derive an approximation algorithm that utilizes an arbitrary approximation algorithm for the well-known Min Set Multicover problem and show that the approximation quality of this algorithm transfers to our algorithm.