  Mader proved that every strongly $k$-connected $n$-vertex digraph contains a
strongly $k$-connected spanning subgraph with at most $2kn - 2k^2$ edges, where
the equality holds for the complete bipartite digraph ${DK}_{k,n-k}$. For dense
strongly $k$-connected digraphs, this upper bound can be significantly
improved. More precisely, we prove that every strongly $k$-connected $n$-vertex
digraph $D$ contains a strongly $k$-connected spanning subgraph with at most
$kn + 800k(k+\overline{\Delta}(D))$ edges, where $\overline{\Delta}(D)$ denotes
the maximum degree of the complement of the underlying undirected graph of a
digraph $D$. Here, the additional term $800k(k+\overline{\Delta}(D))$ is tight
up to multiplicative and additive constants. As a corollary, this implies that
every strongly $k$-connected $n$-vertex semicomplete digraph contains a
strongly $k$-connected spanning subgraph with at most $kn + 800k^2$ edges,
which is essentially optimal since $800k^2$ cannot be reduced to the number
less than $k(k-1)/2$.
  We also prove an analogous result for strongly $k$-arc-connected directed
multigraphs. Both proofs yield polynomial-time algorithms.
