We study the problem of assigning items to agents so as to maximize the \emph{weighted} Nash Social Welfare (NSW) under submodular valuations. The best-known result for the problem is an $O(nw_{\max})$-approximation due to Garg, Husic, Li, Vega, and Vondrak~\cite{GHL23}, where $w_{\max}$ is the maximum weight over all agents. Obtaining a constant approximation algorithm is an open problem in the field that has recently attracted considerable attention. We give the first such algorithm for the problem, thus solving the open problem in the affirmative. Our algorithm is based on the natural Configuration LP for the problem, which was introduced recently by Feng and Li~\cite{FL24} for the additive valuation case. Our rounding algorithm is similar to that of Li \cite{Li25} developed for the unrelated machine scheduling problem to minimize weighted completion time. Roughly speaking, we designate the largest item in each configuration as a large item and the remaining items as small items. So, every agent gets precisely 1 fractional large item in the configuration LP solution. With the rounding algorithm in \cite{Li25}, we can ensure that in the obtained solution, every agent gets precisely 1 large item, and the assignments of small items are negatively correlated.

翻译：本研究探讨在次模估值下分配物品以最大化加权纳什社会福利的问题。该领域目前最优结果为Garg、Husic、Li、Vega和Vondrak提出的$O(nw_{\max})$近似算法，其中$w_{\max}$为所有代理中的最大权重。获得常数近似算法是该领域近期备受关注的重要开放问题。我们针对该问题提出了首个常数近似算法，从而肯定性地解决了该开放问题。我们的算法基于该问题的自然配置线性规划，该规划由Feng和Li近期针对加性估值情形提出。我们的舍入算法与Li为最小化加权完成时间的不相关机器调度问题所设计的方法相似。简而言之，我们将每个配置中的最大物品指定为大物品，其余物品指定为小物品。因此，在配置线性规划解中每个代理恰好获得1个分数化的大物品。通过采用文献中的舍入算法，我们可以确保在最终解中每个代理恰好获得1个大物品，且小物品的分配呈现负相关性。