  Let $\mathcal{W} \subset \mathbb{R}^2$ be a planar polygonal environment
(i.e., a polygon potentially with holes) with a total of $n$ vertices, and let
$A,B$ be two robots, each modeled as an axis-aligned unit square, that can
translate inside $\mathcal{W}$. Given source and target placements
$s_A,t_A,s_B,t_B \in \mathcal{W}$ of $A$ and $B$, respectively, the goal is to
compute a \emph{collision-free motion plan} $\mathbf{\pi}^*$, i.e., a motion
plan that continuously moves $A$ from $s_A$ to $t_A$ and $B$ from $s_B$ to
$t_B$ so that $A$ and $B$ remain inside $\mathcal{W}$ and do not collide with
each other during the motion. Furthermore, if such a plan exists, then we wish
to return a plan that minimizes the sum of the lengths of the paths traversed
by the robots, $\left|\mathbf{\pi}^*\right|$. Given $\mathcal{W},
s_A,t_A,s_B,t_B$ and a parameter $\varepsilon > 0$, we present an
$n^2\varepsilon^{-O(1)} \log n$-time $(1+\varepsilon)$-approximation algorithm
for this problem. We are not aware of any polynomial time algorithm for this
problem, nor do we know whether the problem is NP-Hard. Our result is the first
polynomial-time $(1+\varepsilon)$-approximation algorithm for an optimal motion
planning problem involving two robots moving in a polygonal environment.
