log modular function whether supermodular or submodular

Question

I have a modular function defined as

$$g(X) = \sum_{i \in X} x_i, \quad\text{s.t. } x_i \geq 0$$

Now, I define a function

$$f(X) = \exp(-g(X))$$

As I worked out, this function $f$ is submodular, since $$f(A) + f(B) \geq f(A\cup B) + f(A\cap B)$$

Am I correct ?

Answer 1

I cannot add a comment since I have less than 50 points.

But you can try this. Sumodularity means that incremental increase in the function evaluation on addition of an element to a larger set is less than or equal to the the incremental increase in the function evaluation on addition of the same element to a subset of the larger set.

That is, given $A\subseteq B$ and $x\notin B$.

$f(B\cup {x})-f(B)\leq f(A\cup {x})-f(A)$

Answer 2

Not sure about your result, here is a simple example where $A=\{1,2\}$ and $B=\{2,3\}$, I get

$f(A)=e^{-3},f(B)=e^{-5},f(A\cup B)=e^{-6},f(A\cap B)=e^{-3}$

So overall $f(A)+f(B)\approx 0.05 < 0.14 \approx f(A\cup B)+f(A\cap B)$