Let $\mathcal{R}$ be a transitive and antisymmetric relation defined over a finite set $X$.

For any set $S\subseteq X$ let $\Gamma(S)=\left\{y\in S ,\not \exists x\in S \mid (x,y)\in\mathcal{R}\right\}$. (Thus,$y \in \Gamma(S)$ if it belongs to $S$ and no other element in $S$ "dominates" it.)

Suppose that each element is assigned a weight. This is represented by the function $w:X\to \mathbb{R}^+$.

The problem is to find a subset $S \subseteq X$ to maximize $\sum_{z \in \Gamma(S)}w(z)$.

Is this problem polynomial-time solvable?

Thanks.