Steiner connectivity problems in hypergraphs

We say that a tree T is an S-Steiner tree if S c 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 9-l and some S c V (9-l), whether there is a subhypergraph of 9-l 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 9-l, some r E V (9-l) and some S c V (9-l), 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 9-l and some S c V (9-l), 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 Egervary 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 ISI is fixed.& COPY; 2023 Elsevier B.V. All rights reserved.