Directed paths with few or many colors in colored directed graphs

Given a graph $D=(V(D),A(D))$ and a coloring of $D$, not necessarily a proper coloring of either the arcs or the vertices of $D$, we consider the complexity of finding a path of $D$ from a given vertex $s$ to another given vertex $t$ with as few different colors as possible, and of finding one with as many different colors as possible. We show that the first problem is polynomial-time solvable, and that the second problem is NP-hard