# 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 $$\bigcup_{v \in K_n} L(v) = \{1,2\dots n\}$$?