  Submodular functions are a fundamental object of study in combinatorial
optimization, economics, machine learning, etc. and exhibit a rich
combinatorial structure. Many subclasses of submodular functions have also been
well studied and these subclasses widely vary in their complexity. Our
motivation is to understand the relative complexity of these classes of
functions. Towards this, we consider the question of how well can one class of
submodular functions be approximated by another (simpler) class of submodular
functions. Such approximations naturally allow algorithms designed for the
simpler class to be applied to the bigger class of functions. We prove both
upper and lower bounds on such approximations.
  Our main results are:
  1. General submodular functions can be approximated by cut functions of
directed graphs to a factor of $n^2/4$, which is tight.
  2. General symmetric submodular functions$^{1}$ can be approximated by cut
functions of undirected graphs to a factor of $n-1$, which is tight up to a
constant.
  3. Budgeted additive functions can be approximated by coverage functions to a
factor of $e/(e-1)$, which is tight.
  Here $n$ is the size of the ground set on which the submodular function is
defined.
  We also observe that prior works imply that monotone submodular functions can
be approximated by coverage functions with a factor between $O(\sqrt{n} \log
n)$ and $\Omega(n^{1/3} /\log^2 n) $.
