  We present a simple distributed $\Delta$-approximation algorithm for maximum
weight independent set (MaxIS) in the $\mathsf{CONGEST}$ model which completes
in $O(\texttt{MIS}(G)\cdot \log W)$ rounds, where $\Delta$ is the maximum
degree, $\texttt{MIS}(G)$ is the number of rounds needed to compute a maximal
independent set (MIS) on $G$, and $W$ is the maximum weight of a node. %Whether
our algorithm is randomized or deterministic depends on the \texttt{MIS}
algorithm used as a black-box.
  Plugging in the best known algorithm for MIS gives a randomized solution in
$O(\log n \log W)$ rounds, where $n$ is the number of nodes.
  We also present a deterministic $O(\Delta +\log^* n)$-round algorithm based
on coloring.
  We then show how to use our MaxIS approximation algorithms to compute a
$2$-approximation for maximum weight matching without incurring any additional
round penalty in the $\mathsf{CONGEST}$ model. We use a known reduction for
simulating algorithms on the line graph while incurring congestion, but we show
our algorithm is part of a broad family of \emph{local aggregation algorithms}
for which we describe a mechanism that allows the simulation to run in the
$\mathsf{CONGEST}$ model without an additional overhead.
  Next, we show that for maximum weight matching, relaxing the approximation
factor to ($2+\varepsilon$) allows us to devise a distributed algorithm
requiring $O(\frac{\log \Delta}{\log\log\Delta})$ rounds for any constant
$\varepsilon>0$. For the unweighted case, we can even obtain a
$(1+\varepsilon)$-approximation in this number of rounds. These algorithms are
the first to achieve the provably optimal round complexity with respect to
dependency on $\Delta$.
