  Interpreting three-leaf binary trees or {\em rooted triples} as constraints
yields an entailment relation, whereby binary trees satisfying some rooted
triples must also thus satisfy others, and thence a closure operator, which is
known to be polynomial-time computable. This is extended to inconsistent triple
sets by defining that a triple is entailed by such a set if it is entailed by
any consistent subset of it.
  Determining whether the closure of an inconsistent rooted triple set can be
computed in polynomial time was posed as an open problem in the Isaac Newton
Institute's "Phylogenetics" program in 2007. It appears (as NC4) in a
collection of such open problems maintained by Mike Steel, and it is the last
of that collection's five problems concerning computational complexity to have
remained open. We resolve the complexity of computing this closure, proving
that its decision version is NP-Complete.
  In the process, we also prove that detecting the existence of {\em any}
acyclic B-hyperpath (from specified source to destination) is NP-Complete, in a
significantly narrower special case than the version whose {\em minimization}
problem was recently proven NP-hard by Ritz et al. This implies it is NP-hard
to approximate (our special case of) their minimization problem to within {\em
any} factor.
