We study the problem of approximating maximum Nash social welfare (NSW) when allocating m indivisible items among n asymmetric agents with submodular valuations. The NSW is a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents’ valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with a factor independent of m was known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations before this work.

In this article, we extend our understanding of the NSW problem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations. Both algorithms are simple to understand and involve non-trivial modifications of a greedy repeated matchings approach. Allocations of high-valued items are done separately by un-matching certain items and re-matching them by different processes in both algorithms. We show that these approaches achieve approximation factors of O(n) and O(n log n) for additive and submodular cases, independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item (EF1).

Furthermore, we show that the NSW problem under submodular valuations is strictly harder than all currently known settings with an \(\frac{\mathrm{e}}{\mathrm{e}-1}\) factor of the hardness of approximation, even for constantly many agents. For this case, we provide a different approximation algorithm that achieves a factor of \(\frac{\mathrm{e}}{\mathrm{e}-1}\), hence resolving it completely.

我们研究了在具有子模估值的n 个不对称主体之间分配m 个不可分割的项目时逼近最大纳什社会福利 (NSW) 的问题。这新南威尔士州是一个公认的公平和效率概念，定义为代理人估值的加权几何平均值。对于对称代理和加性（类似）估值函数问题的特殊情况，已经使用针对这些特定设置定制的方法来设计近似算法，并且它们无法扩展到更一般的设置。因此，在这项工作之前，对于具有加性估值的非对称代理或超出加性（类似）估值的对称代理，没有已知具有独立于m的因子的近似算法。

在这篇文章中，我们扩展了对新南威尔士州问题到更一般的设置。我们的主要贡献是具有加法和子模估值的非对称代理的两种近似算法。这两种算法都很容易理解，并且涉及对贪婪重复匹配方法的重要修改。高价值物品的分配是通过两种算法中的不同过程取消匹配某些物品并重新匹配它们来单独完成的。我们证明，这些方法对于加性和子模情况实现了O ( n ) 和O ( n log n )的近似因子，与项目数量无关。对于附加估值，我们的算法输出的分配也实现了最多一项的无嫉妒的公平性（EF1）。

此外，我们表明新南威尔士州子模估值下的问题比所有当前已知的具有近似硬度因子的设置更难，即使对于恒定的多个代理也是如此。对于这种情况，我们提供了一种不同的近似算法，可以实现因子 \(\frac{\mathrm{e}}{\mathrm{e}-1}\)，从而完全解决它。