  A power assignment is an assignment of transmission power to each of the
wireless nodes of a wireless network, so that the induced graph satisfies some
desired properties. The cost of a power assignment is the sum of the assigned
powers. In this paper, we consider the dual power assignment problem, in which
each wireless node is assigned a high- or low-power level, so that the induced
graph is strongly connected and the cost of the assignment is minimized. We
improve the best known approximation ratio from
$\frac{\pi^2}{6}-\frac{1}{36}+\epsilon\thickapprox 1.617$ to
$\frac{11}{7}\thickapprox 1.571$.
  Moreover, we show that the algorithm of Khuller et al. for the strongly
connected spanning subgraph problem, which achieves an approximation ratio of
$1.61$, is $1.522$-approximation algorithm for symmetric directed graphs.
  The innovation of this paper is in achieving these results via utilizing
interesting properties for the existence of a second Hamiltonian cycle.
