  We study the minimum spanning arborescence problem on the complete digraph
$\vec{K}_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of
which is an independent uniform random variable $U^\alpha$ where $\alpha\leq 1$
and $U$ is uniform $[0,1]$. There is also a constraint that the spanning
arborescence $T$ must satisfy $C(T)\leq c_0$. We establish, for a range of
values for $c_0,\alpha$, the asymptotic value of the optimum weight via the
consideration of a dual problem.
