Is there a fast algorithm to detect possible bridge arcs in hyperpaths from set $S$ of nodes to node $t$ in a hypergraph $G$. That is, given a hypergraph $G$, source nodes $S$, and target node $t$, decide whether there exists a hyperarc $A=\langle X, y\rangle \in G$ so that removing $A$ makes target node $t$ unreachable from source nodes $S$.
I am using the usual notation of reachability in hypergraphs as follows: $t$ is reachable from $S$ if $t$ belongs to the minimum set $Z$ satisfying the two following conditions:
$S \subseteq Z$
For all $\langle X, y \rangle \in G$, if $X \subseteq Z$ then $y \in Z$.
Obviously, one can detect bridge arcs in quadratic time by first finding a path from $S$ to $t$ and, for each hyperarc in that path, checking in linear time if $t$ becomes unreachable from $S$ by removing that hypoerarc. I'm looking for anything that might improve on this quadratic upperbound.