  Let $G=(V, E)$ be a simple undirected graph with no isolated vertex. A set
$D_t\subseteq V$ is a total dominating set of $G$ if $(i)$ $D_t$ is a
dominating set, and $(ii)$ the set $D_t$ induces a subgraph with no isolated
vertex. The total dominating set of minimum cardinality is called the minimum
total dominating set, and the size of the minimum total dominating set is
called the total domination number ($\gamma_t(G)$). Given a graph $G$, the
total dominating set (TDS) problem is to find a total dominating set of minimum
cardinality. A Roman dominating function (RDF) on a graph $G$ is a function
$f:V\rightarrow \{0,1,2\}$ such that each vertex $v\in V$ with $f(v)=0$ is
adjacent to at least one vertex $u\in V$ with $f(u)=2$. A RDF $f$ of a graph
$G$ is said to be a total Roman dominating function (TRDF) if the induced
subgraph of $V_1\cup V_2$ does not contain any isolated vertex, where
$V_i=\{u\in V|f(u)=i\}$. Given a graph $G$, the total Roman dominating set
(TRDS) problem is to minimize the weight, $W(f)=\sum_{u\in V} f(u)$, called the
total Roman domination number ($\gamma_{tR}(G)$). In this paper, we are the
first to show that the TRDS problem is NP-complete in unit disk graphs (UDGs).
Furthermore, we propose a $7.17\operatorname{-}$ factor approximation algorithm
for the TDS problem and a $6.03\operatorname{-}$ factor approximation algorithm
for the TRDS problem in geometric unit disk graphs. The running time for both
algorithms is notably bounded by $O(n\log{k})$, where $n$ represents the number
of vertices in the given UDG and $k$ represents the size of the independent set
in (i.e., $D$ and $V_2$ in TDS and TRDS problems, respectively) the given UDG.
