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Maximize function over a set with a transitive and antisymmetric relation

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Problem Detail: 

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

For any set $S\subseteq X$ define $\Gamma(S)=\left\{y\in S \mid \not \exists x\in S . (x,y)\in\mathcal{R}\right\}$. (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?


Answered By : Yuval Filmus

You can compute the maximum weight of an antichain, or more generally the maximum weight of a union of $k$ antichains, by reducing to the maximum flow problem. See for example a technical report by Cong.

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