A function $f:2^V\to \mathbb{R}$ is submodular if for the sets $A$ and $B$ with $A \subset B$ we have that $$f(A) + f(B) \geq f(A\cup B) + f(A\cap B)$$ or equivalently that $$f(A\cup \{v\}) - f(A) \geq f(B\cup \{v\}) - f(B)$$ A function is modular if the function is both submodular and super modular, i.e. the inequalities above both turn into equalities.
It can be shown that we can approximately maximise a submodular function by a greedy algorithm with a worst-case approximation factor of $1-e^{-1}$.
My question is: I've seen studies where they use a greedy algorithm to maximise their function, motivated by the results from maximising submodular functions. However, in their cases they actually define modular functions. Do the same guarantees that apply to solving submodular function maximisation using a greedy algorithm, also apply to modular functions?
To me this is not obvious. Particularly given the second property above, which for submodular functions can be interpreted such that the increase in value by adding $v$ diminishes as the context increases (from $A$ to $B$). But for modular functions this is not the case?
The most common approximate optimality result is for maximizing a nonnegative monotone submodular function subject to a cardinality constraint. Modular functions are submodular, so the result would certainly apply to a nonnegative monotone modular function as well.
However, maximizing (or minimizing) a modular function subject to a cardinality constraint is basically trivial. There is no need for an approximation bound. All modular functions can be written in this form: $$f(A)= f(\emptyset) + \sum_{v \in A} c_v$$
If the constants $c_v$ are not known, then they can be computed by $c_v = f(\{v\})-f(\emptyset)$. So to find the set of size $k$ that maximizes $f$, you just sort the values $c_v$ and add up the $k$ largest.