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 Egerváry 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.

我们说树T是S -Steiner 树，如果$S\subseteq V（\mathrm{时间}）$如果超图可以修剪为S -Steiner 树，则它是S -Steiner 超树。我们证明给定超图，决定是 NP 完全的$\mathcal{H}$还有一些$S\subseteq V（\mathcal{H}）$，是否存在子超图$\mathcal{H}$这是一个S-斯坦纳超树。作为推论，我们对超图中的两个斯坦纳定向问题给出了两个负面结果。首先，我们证明给定一个超图，决定是 NP 完全的$\mathcal{H}$， 一些$r\epsilon V（\mathcal{H}）$还有一些$S\subseteq V（\mathcal{H}）$，该超图是否具有从r可以到达S的每个顶点的方向。其次，我们证明，给定一个超图，决定是 NP 完全的$\mathcal{H}$还有一些$S\subseteq V（\mathcal{H}）$，该超图是否具有S中任意两个顶点相互可达的方向。这回答了 Egerváry 研究小组长期以来的一个悬而未决的问题。我们进一步表明，决定给定超图是否具有良好平衡的方向是 NP 完全的。从积极的一面来看，我们表明，如果终端的数量足够多，那么找到斯坦纳超树的问题和第一方向问题可以在多项式时间内解决$|S|$是固定的。