  We give the first $O(1)$-approximation for the weighted Nash Social Welfare
problem with additive valuations. The approximation ratio we obtain is $e^{1/e}
+ \epsilon \approx 1.445 + \epsilon$, which matches the best known
approximation ratio for the unweighted case \cite{BKV18}.
  Both our algorithm and analysis are simple. We solve a natural configuration
LP for the problem, and obtain the allocation of items to agents using a
randomized version of the Shmoys-Tardos rounding algorithm developed for
unrelated machine scheduling problems. In the analysis, we show that the
approximation ratio of the algorithm is at most the worst gap between the Nash
social welfare of the optimum allocation and that of an EF1 allocation, for an
unweighted Nash Social Welfare instance with identical additive valuations.
This was shown to be at most $e^{1/e} \approx 1.445$ by Barman et al., leading
to our approximation ratio.
