Ratio of two submodular functions is submodular?

Question

Say we had 3 submodular functions $f(X)$, $g(X)$ and $h(X)$ is $\frac{f(X)}{g(X). h(X)}$ submodular as well? What can be said about the submodularity of $\frac{f(X)}{g(X)}$ and $f(X).g(X)$? I understand that the linear combinations of submodular functions are submodular, but what about the above cases? How can we prove or disprove them?

Answer 1

Here is a counterexample. Let $f(x)=\sqrt{x}$. This is submodular. Consider $g(x)=x$. This is linear/submodular. $\frac{f(x)}{g(x)}=x^{-0.5}$ which is not submodular.

In general, only nonnegative linear combinations of submodular functions are submodular. It is not preserved under division or multiplication.