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.
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