We consider minimum equivalent digraph problem, its maximum optimization variant and some non-trivial extensions of these two types of problems motivated by biological and social network applications. We provide $\frac{3}{2}$-approximation algorithms for all the minimization problems and 2-approximation algorithms for all the maximization problems using appropriate primal-dual polytopes. We also show lower bounds on the integrality gap of the polytope to provide some intuition on the final limit of such approaches. Furthermore, we provide APX-hardness result for all those problems even if the length of all simple cycles is bounded by 5.