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.

翻译：我们针对带加性估值的加权纳什社会福利问题给出了首个$O(1)$逼近算法。获得的逼近比为$e^{1/e} + \epsilon \approx 1.445 + \epsilon$，这与无权重情形下的最佳已知逼近比\cite{BKV18}相匹配。我们的算法与分析均简洁明了。通过求解该问题的自然配置线性规划，并采用为无关机器调度问题开发的Shmoys-Tardos随机化舍入算法，得到物品对智能体的分配方案。在分析中，我们证明该算法的逼近比不超过最优分配与EF1分配的纳什社会福利在最坏情况下的差距，该结论针对具有相同加性估值的无权重纳什社会福利实例。Barman等人已证明该差距至多为$e^{1/e} \approx 1.445$，由此得到我们的逼近比。