  In this paper, we consider multiple-antenna $K$-user interference channels
with backhaul collaboration in one side (among the transmitters or among the
receivers) and investigate the trade-off between the rate in the channel versus
the communication load in the backhaul. In this investigation, we focus on a
first order approximation result, where the rate of the wireless channel is
measured by the degrees of freedom (DoF) per user, and the load of the backhaul
is measured by the entropy of backhaul messages per user normalized by $\log$
of transmit power, at high power regimes. This trade-off is fully characterized
for the case of even values of $K$, and approximately characterized for the
case of odd values of $K$, with vanishing approximation gap as $K$ grows. For
full DoF, this result establishes the optimality (approximately) of the most
straightforward scheme, called Centralized Scheme, in which the messages are
collected at one of the nodes, centrally processed, and forwarded back to each
node. In addition, this result shows that the gain of the schemes, relying on
distributed processing, through pairwise communication among the nodes (e.g.,
cooperative alignment) does not scale with the size of the network. For the
converse, we develop a new outer-bound on the trade-off based on splitting the
set of collaborative nodes (transmitters or receivers) into two subsets, and
assuming full cooperation within each group. In continue, we further
investigate the trade-off for the cases, where the backhaul or the wireless
links (interference channel) are not fully connected.
