Complexity of list coloring $K_n$ with $n$ colors

The list coloring problem is, given a set $ L(v)$ colors for each vertex $ v \in G$ , is there a proper vertex coloring, $ c$ , of $ G$ , such that $ c(v) \in L(v), \forall v$ .

I was wondering, for complete graphs $ K_n$ , is list coloring NP-complete? Does this change if $ \bigcup_{v \in K_n} L(v) = \{1,2\dots n\}$ ?