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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.