  We present the first super-linear lower bounds for natural graph problems in
the CONGEST model, answering a long-standing open question.
  Specifically, we show that any exact computation of a minimum vertex cover or
a maximum independent set requires $\Omega(n^2/\log^2{n})$ rounds in the worst
case in the CONGEST model, as well as any algorithm for $\chi$-coloring a
graph, where $\chi$ is the chromatic number of the graph. We further show that
such strong lower bounds are not limited to NP-hard problems, by showing two
simple graph problems in P which require a quadratic and near-quadratic number
of rounds.
  Finally, we address the problem of computing an exact solution to weighted
all-pairs-shortest-paths (APSP), which arguably may be considered as a
candidate for having a super-linear lower bound. We show a simple $\Omega(n)$
lower bound for this problem, which implies a separation between the weighted
and unweighted cases, since the latter is known to have a complexity of
$\Theta(n/\log{n})$. We also formally prove that the standard Alice-Bob
framework is incapable of providing a super-linear lower bound for exact
weighted APSP, whose complexity remains an intriguing open question.
