Range Augmentation Problems in Static Ad-Hoc Wireless Networks

Abstract

Given a set V of n stations, a transmission power cost function c : $V \times V \rightarrowtail {\mathbb R}^{+} \cup {\{+\infty\}}$, an initial power assignment $p_{0} : V \rightarrowtail {\mathbb R}^{+}$, and a connectivity property π, a range augmentation problem consists of finding a minimum power augmentation assignment p such that p _f (·) = p ₀(·) + p(·) satisfies property π. In this paper, we focus on the problem of biconnecting an already existing connected network, to make it resilient to the failure of either a wireless link or a station.For these problems we give a ${\mathcal H}^{2}_{n}$-approximation greedy algorithm (where ${\mathcal H}_{n}$ is the n-th harmonic number) after proving that they are both not approximable within (1 – o(1)) ln n, unless ${\sf NP} \subset {\sf DTIME}(n^{\mathcal{O}({\rm log log} n)})$, even when c is a distance cost function restricted to three power levels, or it is a distance cost function and p ₀ induces a tree network. Moreover, we prove that both problems remain APX-hard even if the initial power assignment is uniform. In this latter scenario, we finally show that any λ-approximation algorithm for the corresponding problem in wired networks, is a 2λ-approximation algorithm for the wireless case.