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：我们给出了具有加法估值的加权纳什社会福利问题的首个 $O(1)$-近似算法。获得的近似比为 $e^{1/e} + \epsilon \approx 1.445 + \epsilon$，这与无权重情况下的最佳已知近似比一致 \cite{BKV18}。我们的算法和分析都较为简洁。我们针对该问题求解了一个自然的配置线性规划，并采用针对无关机调度问题提出的Shmoys-Tardos舍入算法的随机版本得到了物品对智能体的分配。在分析中，我们证明该算法的近似比至多等于无权重纳什社会福利（具有相同加法估值）中，最优分配与EF1分配之间纳什社会福利的最坏差距。Barman等人证明该差距至多为 $e^{1/e} \approx 1.445$，从而得到了我们的近似比。