  A famous conjecture of Tuza states that the minimum number of edges needed to
cover all the triangles in a graph is at most twice the maximum number of
edge-disjoint triangles. This conjecture was couched in a broader setting by
Aharoni and Zerbib who proposed a hypergraph version of this conjecture, and
also studied its implied fractional versions. We establish the fractional
version of the Aharoni-Zerbib conjecture up to lower order terms. Specifically,
we give a factor $t/2+ O(\sqrt{t \log t})$ approximation based on LP rounding
for an algorithmic version of the hypergraph Tur\'{a}n problem (AHTP). The
objective in AHTP is to pick the smallest collection of $(t-1)$-sized subsets
of vertices of an input $t$-uniform hypergraph such that every hyperedge
contains one of these subsets.
  Aharoni and Zerbib also posed whether Tuza's 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 $\mathcal{F}$-free hypergraphs for any finite family
$\mathcal{F}$ of excluded sub-hypergraphs, whose vertex covers must include
almost all the vertices.
  The algorithmic questions arising in the above study can be phrased as
instances of vertex cover on simple hypergraphs, whose hyperedges can pairwise
share at most one vertex. We prove that the trivial factor $t$ approximation
for vertex cover is hard to improve for simple $t$-uniform hypergraphs.
However, for set cover on simple $n$-vertex hypergraphs, the greedy algorithm
achieves a factor $(\ln n)/2$, better than the optimal $\ln n$ factor for
general hypergraphs.
