Steiner connectivity problems in hypergraphs

We say that a tree $T$ is an $S$-Steiner tree if $S \subseteq V(T)$ and a hypergraph is an $S$-Steiner hypertree if it can be trimmed to an $S$-Steiner tree. We prove that it is NP-complete to decide, given a hypergraph $\mathcal{H}$ and some $S \subseteq V(\mathcal{H})$, whether there is a subhypergraph of $\mathcal{H}$ which is an $S$-Steiner hypertree. As corollaries, we give two negative results for two Steiner orientation problems in hypergraphs. Firstly, we show that it is NP-complete to decide, given a hypergraph $\mathcal{H}$, some $r \in V(\mathcal{H})$ and some $S \subseteq V(\mathcal{H})$, whether this hypergraph has an orientation in which every vertex of $S$ is reachable from $r$. Secondly, we show that it is NP-complete to decide, given a hypergraph $\mathcal{H}$ and some $S \subseteq V(\mathcal{H})$, whether this hypergraph has an orientation in which any two vertices in $S$ are mutually reachable from each other. This answers a longstanding open question of the Egerv\'ary Research group. We further show that it is NP-complete to decide if a given hypergraph has a well-balanced orientation. On the positive side, we show that the problem of finding a Steiner hypertree and the first orientation problem can be solved in polynomial time if the number of terminals $|S|$ is fixed.