On minimizing the total power of <Emphasis Type="Italic">k</Emphasis>-strongly connected wireless networks

Given a wireless network, we want to assign each node a transmission power, which will enable transmission between any two nodes (via other nodes). Moreover, due to possible faults, we want to have at least k vertex-disjoint paths from any node to any other, where k is some fixed integer, depending on the reliability of the nodes. The goal is to achieve this directed k-strong connectivity with a minimal overall power assignment. The problem is NP-Hard for any k ≥ 1 already for planar networks. Here we first present an optimal power assignment for uniformly spaced nodes on a line for any k ≥ 1. We also prove a number of useful properties of power assignment which are also of independent interest. Based on it, we design an approximation algorithm for linear radio networks with factor min{2,(\frac \Updelta d)^a },\hbox{min}\left\{2,\left(\frac {\Updelta} {\delta}\right)^\alpha \right\}, where Δ and δ are the maximal and minimal distances between adjacent nodes respectively and parameter α ≥ 1 being the distance-power gradient. We then extend it to the weighted version. Finally, we develop an approximation algorithm with factor O(k ²), for planar case, which is, to the best of our knowledge, the first non-trivial result for this problem.