  We investigate problems addressing combined connectivity augmentation and
orientations settings. We give a polynomial-time 6-approximation algorithm for
finding a minimum cost subgraph of an undirected graph $G$ that admits an
orientation covering a nonnegative crossing $G$-supermodular demand function,
as defined by Frank. An important example is $(k,\ell)$-edge-connectivity, a
common generalization of global and rooted edge-connectivity.
  Our algorithm is based on a non-standard application of the iterative
rounding method. We observe that the standard linear program with cut
constraints is not amenable and use an alternative linear program with
partition and co-partition constraints instead. The proof requires a new type
of uncrossing technique on partitions and co-partitions.
  We also consider the problem setting when the cost of an edge can be
different for the two possible orientations. The problem becomes substantially
more difficult already for the simpler requirement of $k$-edge-connectivity.
Khanna, Naor, and Shepherd showed that the integrality gap of the natural
linear program is at most $4$ when $k=1$ and conjectured that it is constant
for all fixed $k$. We disprove this conjecture by showing an $\Omega(|V|)$
integrality gap even when $k=2$.
