Minimum Path cover in a Directed Acyclic Graph
This answer shows this problem is NP-hard. Further research is needed to determine whether it belongs to APX or it is APX-hard.

Let $G$ be a complete directed acyclic graph, i.e., you can name the vertices $1,2,\ldots,n$ and there is an edge $(i,j)$ for all $i&lt;j$. Let $D'=D$, and $k$ is equal to half of the sum of the weights of all vertices. Now there exist 2 vertex-disjoint paths satisfying your conditions if and only if the weights can be partitioned into two subsets both with sum $k$, which is exactly the partition problem. Hence, your problem is NP-hard.