In an instance of the weighted Nash Social Welfare problem, we are given a set of $m$ indivisible items, $\mathscr{G}$, and $n$ agents, $\mathscr{A}$, where each agent $i \in \mathscr{A}$ has a valuation $v_{ij}\geq 0$ for each item $j\in \mathscr{G}$. In addition, every agent $i$ has a non-negative weight $w_i$ such that the weights collectively sum up to $1$. The goal is to find an assignment $\sigma:\mathscr{G}\rightarrow \mathscr{A}$ that maximizes $\prod_{i\in \mathscr{A}} \left(\sum_{j\in \sigma^{-1}(i)} v_{ij}\right)^{w_i}$, the product of the weighted valuations of the players. When all the weights equal $\frac1n$, the problem reduces to the classical Nash Social Welfare problem, which has recently received much attention. In this work, we present a $5\cdot\exp\left(2\cdot D_{\text{KL}}(\mathbf{w}\, ||\, \frac{\vec{\mathbf{1}}}{n})\right) = 5\cdot\exp\left(2\log{n} + 2\sum_{i=1}^n w_i \log{w_i}\right)$-approximation algorithm for the weighted Nash Social Welfare problem, where $D_{\text{KL}}(\mathbf{w}\, ||\, \frac{\vec{\mathbf{1}}}{n})$ denotes the KL-divergence between the distribution induced by $\mathbf{w}$ and the uniform distribution on $[n]$. We show a novel connection between the convex programming relaxations for the unweighted variant of Nash Social Welfare presented in \cite{cole2017convex, anari2017nash}, and generalize the programs to two different mathematical programs for the weighted case. The first program is convex and is necessary for computational efficiency, while the second program is a non-convex relaxation that can be rounded efficiently. The approximation factor derives from the difference in the objective values of the convex and non-convex relaxation.

翻译：在权重纳什社会福利问题的一个实例中，给定一组 $m$ 个不可分割物品 $\mathscr{G}$ 和 $n$ 个智能体 $\mathscr{A}$，其中每个智能体 $i \in \mathscr{A}$ 对每个物品 $j \in \mathscr{G}$ 具有非负估值 $v_{ij}\geq 0$。此外，每个智能体 $i$ 有一个非负权重 $w_i$，且所有权重之和为 $1$。目标是为每个物品指派一个智能体 $\sigma:\mathscr{G}\rightarrow \mathscr{A}$，以最大化 $\prod_{i\in \mathscr{A}} \left(\sum_{j\in \sigma^{-1}(i)} v_{ij}\right)^{w_i}$，即智能体加权估值的乘积。当所有权重均等于 $\frac1n$ 时，问题退化为经典的纳什社会福利问题，该问题近期备受关注。本文针对权重纳什社会福利问题，提出一个 $5\cdot\exp\left(2\cdot D_{\text{KL}}(\mathbf{w}\, ||\, \frac{\vec{\mathbf{1}}}{n})\right) = 5\cdot\exp\left(2\log{n} + 2\sum_{i=1}^n w_i \log{w_i}\right)$-近似算法，其中 $D_{\text{KL}}(\mathbf{w}\, ||\, \frac{\vec{\mathbf{1}}}{n})$ 表示由权重 $\mathbf{w}$ 诱导的分布与 $[n]$ 上均匀分布之间的 KL 散度。我们揭示了文献 \cite{cole2017convex, anari2017nash} 中无权重纳什社会福利问题的凸规划松弛之间的一种新颖联系，并将这些规划推广到加权情况下的两种不同的数学规划。第一个规划是凸的，对计算效率至关重要；第二个规划是非凸松弛，可被高效舍入。近似因子源自凸与非凸规划目标函数值的差异。