  We present improved results for approximating maximum-weight independent set
($\MaxIS$) in the CONGEST and LOCAL models of distributed computing. Given an
input graph, let $n$ and $\Delta$ be the number of nodes and maximum degree,
respectively, and let $\MIS(n,\Delta)$ be the the running time of finding a
\emph{maximal} independent set ($\MIS$) in the CONGEST model. Bar-Yehuda et al.
[PODC 2017] showed that there is an algorithm in the CONGEST model that finds a
$\Delta$-approximation for $\MaxIS$ in $O(\MIS(n,\Delta)\log W)$ rounds, where
$W$ is the maximum weight of a node in the graph, which can be as high as
$\poly (n)$. Whether their algorithm is deterministic or randomized depends on
the $\MIS$ algorithm that is used as a black-box.
  Our main result in this work is a randomized $(\poly(\log\log
n)/\epsilon)$-round algorithm that finds, with high probability, a
$(1+\epsilon)\Delta$-approximation for $\MaxIS$ in the CONGEST model. That is,
by sacrificing only a tiny fraction of the approximation guarantee, we achieve
an \emph{exponential} speed-up in the running time over the previous best known
result. Due to a lower bound of $\Omega(\sqrt{\log n/\log \log n})$ that was
given by Kuhn, Moscibroda and Wattenhofer [JACM, 2016] on the number of rounds
for any (possibly randomized) algorithm that finds a maximal independent set
(even in the LOCAL model) this result implies that finding a
$(1+\epsilon)\Delta$-approximation for $\MaxIS$ is exponentially easier than
$\MIS$.
