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 $(poly(loglog n)/epsilon)$-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 $MIS$.
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