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.
翻译:我们研究的是以公平和经济高效的方式在代理人之间分配不可分割货物的问题。在这方面,纳什社会福利被定义为代理人对其分配的捆绑站进行估价的几何平均值,作为衡量分配公平程度的一项基本措施。侧重于代理人估价具有二进制边际的事例,我们从本质上为(估计)在两个最普通的无补充估值类别(即二进制的XOS和二进制的次级补充性估值)中最大限度地实现纳什社会福利而得出了紧凑的结果。对于XOS的二进制估值,我们开发了一种混合时间算法,该算法在标准价值-甲级模型中为最佳纳什社会福利找到一个常数(具体为288美元)近似值。我们的算法所计算的分配也实现了社会福利的常数近似近似值和群体最大分享保证。这些结果意味着,在二进制的XOS估值中,一定存在一种同时满足多重(近似)公平和效率标准的分配。在X进制的亚进制的X级估值中,我们对X进制的排序结果加以补充,从而证明X进进进进制的社会福利评估在X的硬的亚性估价。