  We consider the problem of enumerating all minimal transversals (also called
minimal hitting sets) of a hypergraph $\mathcal{H}$. An equivalent formulation
of this problem known as the \emph{transversal hypergraph} problem (or
\emph{hypergraph dualization} problem) is to decide, given two hypergraphs,
whether one corresponds to the set of minimal transversals of the other. The
existence of a polynomial time algorithm to solve this problem is a long
standing open question. In \cite{fredman_complexity_1996}, the authors present
the first sub-exponential algorithm to solve the transversal hypergraph problem
which runs in quasi-polynomial time, making it unlikely that the problem is
(co)NP-complete.
  In this paper, we show that when one of the two hypergraphs is of bounded
VC-dimension, the transversal hypergraph problem can be solved in polynomial
time, or equivalently that if $\mathcal{H}$ is a hypergraph of bounded
VC-dimension, then there exists an incremental polynomial time algorithm to
enumerate its minimal transversals. This result generalizes most of the
previously known polynomial cases in the literature since they almost all
consider classes of hypergraphs of bounded VC-dimension. As a consequence, the
hypergraph transversal problem is solvable in polynomial time for any class of
hypergraphs closed under partial subhypergraphs. We also show that the proposed
algorithm runs in quasi-polynomial time in general hypergraphs and runs in
polynomial time if the conformality of the hypergraph is bounded, which is one
of the few known polynomial cases where the VC-dimension is unbounded.
