  A real-valued set function is (additively) approximately submodular if it
satisfies the submodularity conditions with an additive error. Approximate
submodularity arises in many settings, especially in machine learning, where
the function evaluation might not be exact. In this paper we study how close
such approximately submodular functions are to truly submodular functions.
  We show that an approximately submodular function defined on a ground set of
$n$ elements is $O(n^2)$ pointwise-close to a submodular function. This result
also provides an algorithmic tool that can be used to adapt existing submodular
optimization algorithms to approximately submodular functions. To complement,
we show an $\Omega(\sqrt{n})$ lower bound on the distance to submodularity.
  These results stand in contrast to the case of approximate modularity, where
the distance to modularity is a constant, and approximate convexity, where the
distance to convexity is logarithmic.
