Abstract
We say that a tree T is an S-Steiner tree if S⊆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 H and some S⊆V(H), whether there is a subhypergraph of 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 H, some r∈V(H) and some S⊆V(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 H and some S⊆V(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.
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