  We consider classes of objective functions of cardinality constrained
maximization problems for which the greedy algorithm guarantees a constant
approximation. We propose the new class of $\gamma$-$\alpha$-augmentable
functions and prove that it encompasses several important subclasses, such as
functions of bounded submodularity ratio, $\alpha$-augmentable functions, and
weighted rank functions of an independence system of bounded rank quotient - as
well as additional objective functions for which the greedy algorithm yields an
approximation. For this general class of functions, we show a tight bound of
$\frac{\alpha}{\gamma}\cdot\frac{\mathrm{e}^\alpha}{\mathrm{e}^\alpha-1}$ on
the approximation ratio of the greedy algorithm that tightly interpolates
between bounds from the literature for functions of bounded submodularity ratio
and for $\alpha$-augmentable functions. In paritcular, as a by-product, we
close a gap left in [Math.Prog., 2020] by obtaining a tight lower bound for
$\alpha$-augmentable functions for all $\alpha\geq1$. For weighted rank
functions of independence systems, our tight bound becomes
$\frac{\alpha}{\gamma}$, which recovers the known bound of $1/q$ for
independence systems of rank quotient at least $q$.
