Are Gross Substitutes a Substitute for Submodular Valuations?

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.