  A mixed dominating set of a graph $G = (V, E)$ is a mixed set $D$ of vertices
and edges, such that for every edge or vertex, if it is not in $D$, then it is
adjacent or incident to at least one vertex or edge in $D$. The mixed
domination problem is to find a mixed dominating set with a minimum
cardinality. It has applications in system control and some other scenarios and
it is $NP$-hard to compute an optimal solution. This paper studies
approximation algorithms and hardness of the weighted mixed dominating set
problem. The weighted version is a generalization of the unweighted version,
where all vertices are assigned the same nonnegative weight $w_v$ and all edges
are assigned the same nonnegative weight $w_e$, and the question is to find a
mixed dominating set with a minimum total weight. Although the mixed dominating
set problem has a simple 2-approximation algorithm, few approximation results
for the weighted version are known. The main contributions of this paper
include: [1.] for $w_e\geq w_v$, a 2-approximation algorithm; [2.] for $w_e\geq
2w_v$, inapproximability within ratio 1.3606 unless $P=NP$ and within ratio 2
under UGC; [3.] for $2w_v > w_e\geq w_v$, inapproximability within ratio 1.1803
unless $P=NP$ and within ratio 1.5 under UGC; [4.] for $w_e< w_v$,
inapproximability within ratio $(1-\epsilon)\ln |V|$ unless $P=NP$ for any
$\epsilon >0$.
