  Considering a communication topology of a wireless network modeled by a graph
where an edge exists between two nodes if they are within each other's
communication range. A subset $U$ of nodes is a dominating set if each node is
either in $U$ or adjacent to some node in $U$. Assume each node has a disparate
communication range and is associated with a positive weight, we present a
randomized algorithm to find a min-weight dominating set. Considering any
orientation of the graph where an arc $\overrightarrow{uv}$ exists if the node
$v$ lies in $u$'s communication range. A subset $U$ of nodes is a strongly
dominating set if every node except $U$ has both in-neighbor(s) and
out-neighbor(s) in $U$. We present a polynomial-time algorithm to find a
strongly dominating set of size at most $(2+\epsilon)$ times of the optimum. We
also investigate another related problem called $K$-Coverage. Given are a set
${\cal D}$ of disks with positive weight and a set ${\cal P}$ of nodes. Assume
all input nodes lie below a horizontal line $l$ and all input disks lie above
this line $l$ in the plane. The objective is to find a min-weight subset ${\cal
D}'\subseteq {\cal D}$ of disks such that each node is covered at least $K$
disks in ${\cal D}'$. We propose a novel two-approximation algorithm for this
problem.
