  We consider the problem of maximizing non-negative non-decreasing set
functions. Although most of the recent work focus on exploiting submodularity,
it turns out that several objectives we encounter in practice are not
submodular. Nonetheless, often we leverage the greedy algorithms used in
submodular functions to determine a solution to the non-submodular functions.
Hereafter, we propose to address the original problem by \emph{approximating}
the non-submodular function and analyze the incurred error, as well as the
performance trade-offs. To quantify the approximation error, we introduce a
novel concept of $\delta$-approximation of a function, which we used to define
the space of submodular functions that lie within an approximation error. We
provide necessary conditions on the existence of such $\delta$-approximation
functions, which might not be unique. Consequently, we characterize this
subspace which we refer to as \emph{region of submodularity}. Furthermore,
submodular functions are known to lead to different sub-optimality guarantees,
so we generalize those dependencies upon a $\delta$-approximation into the
notion of \emph{greedy curvature}. Finally, we used this latter notion to
simplify some of the existing results and efficiently (i.e., linear complexity)
determine tightened bounds on the sub-optimality guarantees using objective
functions commonly used in practical setups and validate them using real data.
