  The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced
by Chartrand et al. in 1984, which is a nice generalization of the classical
connectivity. Recently, as a natural counterpart, Li et al. proposed the
concept of generalized edge-connectivity for a graph. In this paper, we
determine the computational complexity of the generalized connectivity and
generalized edge-connectivity of a graph. Two conjectures are also proved to be
true.
