  Given a positive integer $k$, a $k$-dominating set in a graph $G$ is a set of
vertices such that every vertex not in the set has at least $k$ neighbors in
the set. A total $k$-dominating set, also known as a $k$-tuple total dominating
set, is a set of vertices such that every vertex of the graph has at least $k$
neighbors in the set. The problems of finding the minimum size of a
$k$-dominating, respectively total $k$-dominating set, in a given graph, are
referred to as $k$-domination, respectively total $k$-domination. These
generalizations of the classical domination and total domination problems are
known to be NP-hard in the class of chordal graphs, and, more specifically,
even in the classes of split graphs (both problems) and undirected path graphs
(in the case of total $k$-domination). On the other hand, it follows from
recent work of Kang et al.~(2017) that these two families of problems are
solvable in time $\mathcal{O}(|V(G)|^{6k+4})$ in the class of interval graphs.
We develop faster algorithms for $k$-domination and total $k$-domination in the
class of proper interval graphs, by means of reduction to a single shortest
path computation in a derived directed acyclic graph with
$\mathcal{O}(|V(G)|^{2k})$ nodes and $\mathcal{O}(|V(G)|^{4k})$ arcs. We show
that a suitable implementation, which avoids constructing all arcs of the
digraph, leads to a running time of $\mathcal{O}(|V(G)|^{3k})$. The algorithms
are also applicable to the weighted case.
