Fault-tolerant power assignment and backbone in wireless networks

Given a wireless network, we need to assign transmission power to each of the nodes, that will enable communication between any two nodes (via other nodes). Moreover, due to possible faults, we would like to have at least k vertex-disjoint paths from any node to any other node, where k is some fixed integer, depending on the reliability of the nodes. The goal is to achieve this directed k-connectivity with minimal overall power consumption. The problem is NP-Hard for any k ges 1, already for planar networks. Here we develop an O(k)-approximation algorithm for the planar case. We also address the problem of constructing a connected backbone, for which we present an efficient constant-factor approximation algorithm