  A fundamental class of problems in wireless communication is concerned with
the assignment of suitable transmission powers to wireless devices/stations
such that the resulting communication graph satisfies certain desired
properties and the overall energy consumed is minimized. Many concrete
communication tasks in a wireless network like broadcast, multicast,
point-to-point routing, creation of a communication backbone, etc. can be
regarded as such a power assignment problem.
  This paper considers several problems of that kind; for example one problem
studied before in \cite{Carrots, Bilo} aims to select and assign powers to $k$
of the stations such that all other stations are within reach of at least one
of the selected stations. We improve the running time for obtaining a
$(1+\epsilon)$-approximate solution for this problem from
$n^{((\alpha/\epsilon)^{O(d)})}$ as reported by Bilo et al. (\cite{Bilo}) to
$O(n+ {(\frac{k^{2d+1}}{\epsilon^d})}^{\min{\{2k, (\alpha/\epsilon)^{O(d)}
\}}})$ that is, we obtain a running time that is \emph{linear} in the network
size. Further results include a constant approximation algorithm for the TSP
problem under squared (non-metric!) edge costs, which can be employed to
implement a novel data aggregation protocol, as well as efficient schemes to
perform $k$-hop multicasts.
