  Experimental design is a classical statistics problem and its aim is to
estimate an unknown $m$-dimensional vector $\beta$ from linear measurements
where a Gaussian noise is introduced in each measurement. For the combinatorial
experimental design problem, the goal is to pick $k$ out of the given $n$
experiments so as to make the most accurate estimate of the unknown parameters,
denoted as $\hat{\beta}$. In this paper, we will study one of the most robust
measures of error estimation - $D$-optimality criterion, which corresponds to
minimizing the volume of the confidence ellipsoid for the estimation error
$\beta-\hat{\beta}$. The problem gives rise to two natural variants depending
on whether repetitions of experiments are allowed or not. We first propose an
approximation algorithm with a $\frac1e$-approximation for the $D$-optimal
design problem with and without repetitions, giving the first constant factor
approximation for the problem. We then analyze another sampling approximation
algorithm and prove that it is $(1-\epsilon)$-approximation if $k\geq
\frac{4m}{\epsilon}+\frac{12}{\epsilon^2}\log(\frac{1}{\epsilon})$ for any
$\epsilon \in (0,1)$. Finally, for $D$-optimal design with repetitions, we
study a different algorithm proposed by literature and show that it can improve
this asymptotic approximation ratio.
