k-fault resistance in wireless ad-hoc 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 <i>k</i> vertex-disjoint paths from any node to any other, where <i>k</i> is some fixed integer, depending on the reliability of the nodes. The goal is to achieve this <i>directed</i> <i>k</i>-connectivity with a minimal overall power assignment. The problem is <b>NP-Hard</b> for any <i>k ≥ 1</i> already for planar networks. Here we first present an optimal power assignment for uniformly spaced nodes on a line for any <i>k ≥ 1</i>. Based on it, we design an approximation algorithm for linear radio networks with factor <i>min(2,(Δ/δ)<sup>α</sup>)</i>, where <i>Δ</i> and <i>δ</i> are the maximal and minimal distances between adjacent nodes respectively and parameter <i>α ≥ 1</i> being the distance-power gradient. We then extend it to the weighted version. Finally, we develop an approximation algorithm with factor <i>O(k<sup>2</sup>)</i>, for planar case, which is, to the best of our knowledge, the first non-trivial result for this problem.

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