  We study the problem of computing a minimum equivalent digraph (also known as
the problem of computing a strong transitive reduction) and its maximum
objective function variant, with two types of extensions. First, we allow to
declare a set $D\subset E$ and require that a valid solution $A$ satisfies
$D\subset A$ (it is sometimes called transitive reduction problem). In the
second extension (called $p$-ary transitive reduction), we have integer edge
labeling and we view two paths as equivalent if they have the same beginning,
ending and the sum of labels modulo $p$. A solution $A\subseteq E$ is valid if
it gives an equivalent path for every original path. For all problems we
establish the following: polynomial time minimization of $|A|$ within ratio
1.5, maximization of $|E-A|$ within ratio 2, MAX-SNP hardness even of the
length of simple cycles is limited to 5. Furthermore, we believe that the
combinatorial technique behind the approximation algorithm for the minimization
version might be of interest to other graph connectivity problems as well.
