  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.
