Before revision:  AHTP admits a trivial factor t approximation algorithm as it can be cast as an instance of vertex cover on a structured t-uniform hypergraph that is a ``blown-up'' version of H. Our main result is an approximation algorithm with ratio \frac{t simple %DIFAUXCMD hypergraph with hyperedges intersecting in at most one element .
After revision:  AHTP admits a trivial factor t approximation algorithm as it can be cast as an instance of vertex cover on a structured t-uniform hypergraph that is a ``blown-up'' version of H. Our main result is an approximation algorithm with ratio \frac{t simple %DIFAUXCMD conjecture and its hypergraph versions could follow from non-trivial duality gaps between vertex covers and matchings on hypergraphs that exclude certain sub-hypergraphs, for instance, a "tent" structure that cannot occur in the incidence of triangles and edges. We give a strong negative answer to this question, by exhibiting tent-free hypergraphs, and indeed \mathcal{F at most one element .