  This paper studies lower bounds for fundamental optimization problems in the
CONGEST model. We show that solving problems exactly in this model can be a
hard task, by providing $\tilde{\Omega}(n^2)$ lower bounds for cornerstone
problems, such as minimum dominating set (MDS), Hamiltonian path, Steiner tree
and max-cut. These are almost tight, since all of these problems can be solved
optimally in $O(n^2)$ rounds. Moreover, we show that even in bounded-degree
graphs and even in simple graphs with maximum degree 5 and logarithmic
diameter, it holds that various tasks, such as finding a maximum independent
set (MaxIS) or a minimum vertex cover, are still difficult, requiring a
near-tight number of $\tilde{\Omega}(n)$ rounds.
  Furthermore, we show that in some cases even approximations are difficult, by
providing an $\tilde{\Omega}(n^2)$ lower bound for a
$(7/8+\epsilon)$-approximation for MaxIS, and a nearly-linear lower bound for
an $O(\log{n})$-approximation for the $k$-MDS problem for any constant $k \geq
2$, as well as for several variants of the Steiner tree problem.
  Our lower bounds are based on a rich variety of constructions that leverage
novel observations, and reductions among problems that are specialized for the
CONGEST model. However, for several additional approximation problems, as well
as for exact computation of some central problems in $P$, such as maximum
matching and max flow, we show that such constructions cannot be designed, by
which we exemplify some limitations of this framework.
