  Let $\mathcal{F}$ be a set consisting of $n$ real vectors of dimension $m
\leq n$. For any saturated, i.e., $m$-element, subset $\mathcal{S}$ of
$\mathcal{F}$, let $\mathrm{vol}(\mathcal{S})$ be the volume of the
parallelotope formed by the vectors of $\mathcal{S}$. A set $\mathcal{S}^*$ is
called a $D$-optimal saturated subset of $\mathcal{F}$, if it maximizes
$\mathrm{vol}(\mathcal{S})$ among all saturated subsets of $\mathcal{F}$. In
this paper, we propose two greedy heuristics for the construction of saturated
subsets performing well with respect to the criterion of $D$-optimality: an
improvement of the method suggested by Galil and Kiefer for the initiation of
$D$-optimal experimental design algorithms, and a modification of the
Kumar-Yildirim method, the original version of which was proposed for the
initiation of the minimum-volume enclosing ellipsoid algorithms. We provide
geometric and analytic insights into the two methods, and compare them to the
commonly used random and regularized greedy heuristics. We also suggest
variants of the greedy methods for a large set $\mathcal{F}$, for the
construction of $D$-efficient non-saturated subsets, and for alternative
optimality criteria.
