I am trying to understand whether a monotone and subadditive function $f(S), S \subseteq 2^{[n]}$ and $ f : 2^{[n]} \rightarrow R_{\geq0}$ can be represented using an additive function $\hat{f}(S) = \sum_{i \in S} w_i$. In other words, is it possible to approximate $f$ for all $S$ by finding the value of the weights $w_i$?
The best I can do is the following : $\hat{f} = \sum_{i \in S} \frac{f({i})}{n}$ which can be shown to be $\frac{f(S)}{n} \leq \hat{f}(S) \leq f(S)$ i.e an $n-$approximation. Note that I just need to argue about existence of such a function and not the construction. There are known results on how this is impossible to do in subexponential queries using an oracle that gives you access to $f$ and queries are of the form $f(S)$ for any $S$. I am looking for lower bounds assuming infinite computational power. Is it possible to show that this is tight and we cannot hope to have a better function?
My intuition is that a subadditive function is still pretty linear and a bad instance would be when the values of all sets of fixed size (say $k$) vary wildly rather than being a constant/additive/linearly.
Any reference about representing function using additive linear functions will be very helpful!
