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. In particular, the largest gap known between Submodular and GS valuations was some constant ratio smaller than 2. 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. However, whether GS approximates general submodular valuations within a poly-logarithmic factor remains open, even in the special case of {\em concave of GS} valuations (a subclass of Submodular containing budget additive). For {\em concave of Rado} valuations (Rado is a significant subclass of GS, containing, e.g., weighted matroid rank functions and OXS), we show approximability by GS within an $O(\log^2m)$ factor.

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总替代量是否替代了亚模估价？

总替代品（GS）集的功能类别在经济学和计算机科学中扮演着重要角色。GS属于Lehmann，Lehmann和Nisan引入的{\ em补充免费}评估等级，以及其他主要类别：$ GS \ subsetneq次模块\ subsetneq XOS \ subsetneq Subadditive $。GS类在定义和与其他类的关系上一直比对等类更神秘。例如，虽然众所周知，子模块类，XOS类和子加性类（逐点）之间有多接近，但GS对这些类的近似性仍未解决。特别地，在次模块和GS估值之间已知的最大差距是一些小于2的恒定比率。我们的主要结果是，存在一个子模块化估值（也可以作为预算加总），在GS不能以高于$ \ Omega（\ frac {\ log m} {\ log \ log m}）$的比率进行近似估算时，其中$ m $是项目数。在途中，我们发现了一个新的对称化操作，该操作可以保护GS，这可能是与您个人利益相关的。我们表明，我们的主要结果在预算添加剂估值方面是严格的。但是，即使在{\ em凹入GS}估值（包含预算添加剂的Submodular的子类）的特殊情况下，GS在多对数因子内是否近似于一般的子模块估值仍然是未知的。对于{\ Rado的凹面}估值（Rado是GS的重要子类，包含例如加权拟阵秩函数和OXS），我们显示了GS在$ O（\ log ^ 2m）$因子内的近似性。