Clustering without replication: approximation and inapproximability

We consider the problem of clustering the nodes of directed acyclic graphs when the replication of logic is not allowed. We show that the problem is {\bf NP-hard} even when the vertices of the DAG are unweighted and the cluster capacity is $2$. Moreover, when the vertices of the DAG are weighted, the problem does not admit a $(2-\epsilon)$-approximation algorithm for each $\epsilon >0$, unless {\bf P=NP}. On the positive side, we show that in case the vertices of the DAG are unweighted and $M=2$, the problem admits a $2$-approximation algorithm. Finally, we present some cases when the problem can be solved in polynomial time.