The class of gross substitutes (GS) set functions plays a central role in Economics and Computer Science. GS belongs to the hierarchy of {\em complement free} valuations introduced by Lehmann, Lehmann and Nisan, along with other prominent classes: $GS \subsetneq Submodular \subsetneq XOS \subsetneq Subadditive$. The GS class has always been more enigmatic than its counterpart classes, both in its definition and in its relation to the other classes. For example, while it is well understood how closely the Submodular, XOS and Subadditive classes (point-wise) approximate one another, approximability of these classes by GS remained wide open. Our main result is the existence of a submodular valuation (one that is also budget additive) that cannot be approximated by GS within a ratio better than $\Omega(\frac{\log m}{\log\log m})$, where $m$ is the number of items. En route, we uncover a new symmetrization operation that preserves GS, which may be of independent interest. We show that our main result is tight with respect to budget additive valuations. Additionally, for a class of submodular functions that we refer to as {\em concave of Rado} valuations (this class greatly extends budget additive valuations), we show approximability by GS within an $O(\log^2m)$ factor.

翻译：在经济学和计算机科学中, 毛替代品( GS) 设置的功能类别在经济学和计算机科学中发挥着核心作用。 GS属于列曼、 莱曼和尼桑 和其他著名类别( $GS\ subsetneq Submodual Submodual \ subsetneq XOS \ subsetneq Subaddivisive$) 推出的 Excial 和计算机科学中。 GS 类别在定义上和与其他类别的关系上, 总是比对应类别更复杂。 例如, GS 属于子模块、 XOS 和 Subadditive 类( 点) 的等级, 彼此接近, 这些类的相似性仍然很开放。 我们的主要结果是, GS 的等级值比值不能比 $Omega( orcocolog mlog) 更接近 。 我们发现一个新的 IMDISG 和 RLO 的等级值, 我们的排序比值代表了我们预算的等级值的等级值。