We study the problem of allocating indivisible goods among agents in a fair and economically efficient manner. In this context, the Nash social welfare--defined as the geometric mean of agents' valuations for their assigned bundles--stands as a fundamental measure that quantifies the extent of fairness of an allocation. Focusing on instances in which the agents' valuations have binary marginals, we develop essentially tight results for (approximately) maximizing Nash social welfare under two of the most general classes of complement-free valuations, i.e., under binary XOS and binary subadditive valuations. For binary XOS valuations, we develop a polynomial-time algorithm that finds a constant-factor (specifically 288) approximation for the optimal Nash social welfare, in the standard value-oracle model. The allocations computed by our algorithm also achieve constant-factor approximation for social welfare and the groupwise maximin share guarantee. These results imply that--in the case of binary XOS valuations--there necessarily exists an allocation that simultaneously satisfies multiple (approximate) fairness and efficiency criteria. We complement the algorithmic result by proving that Nash social welfare maximization is APX-hard under binary XOS valuations. Furthermore, this work establishes an interesting separation between the binary XOS and binary subadditive settings. In particular, we prove that an exponential number of value queries are necessarily required to obtain even a sub-linear approximation for Nash social welfare under binary subadditive valuations.

中文翻译：

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在二元 XOS 和二元次加法估值下近似纳什社会福利

我们研究了以公平和经济有效的方式在代理之间分配不可分割的商品的问题。在这种情况下，纳什社会福利——定义为代理人对其分配的束的估值的几何平均值——是量化分配公平程度的基本衡量标准。关注代理的估值具有二元边际的情况，我们在两个最一般的无补充估值类别下（即，在二元 XOS 和二元次可加估值下）为（大约）最大化纳什社会福利开发了本质上严格的结果。对于二元 XOS 估值，我们开发了一个多项式时间算法，该算法在标准价值预言模型中找到最优 Nash 社会福利的常数因子（特别是 288）近似值。由我们的算法计算的分配也实现了社会福利的常数因子近似和分组最大化份额保证。这些结果意味着——在二元 XOS 估值的情况下——必然存在同时满足多个（近似）公平和效率标准的分配。我们通过证明 Nash 社会福利最大化在二进制 XOS 估值下是 APX 难的来补充算法结果。此外，这项工作在二进制 XOS 和二进制 subadditive 设置之间建立了一个有趣的分离。特别是，我们证明了在二元次可加估值下，即使是 Nash 社会福利的次线性近似，也需要指数数量的值查询。