MINIMUM COST TOPOLOGICAL ORDERING

We are given an $n$ vertex directed graph $G=(V,E)$ and also given a cost function $c:V\times [n]\to \mathbb{R}$. Consider a topological ordering of the vertices, $v_1,\ldots,v_n$, the cost of the ordering is $\sum_{i=1}^n c(v_i,i)$. We shall prove that finding the minimum cost topological ordering is $\mathrm{NP}$-hard.