Let's say I can define a group $G$ acting on a set of combinatorial objects $X$ and I have a function $f: X \to \Bbb{N}$ that I want to find a minimum of in $X$.
Is there a polynomial time algorithm using thes above tools to find a minimum of $X$ starting with an input that is a maximum of $X$? I mean the running time is $O(P(n))$ where $P$ is some polynomial and $n = f(x_0)$ is a maximum?
If the answer is no or idk, is there if we add more requirements to $G$ and $f$?