What is the complexity of the following problem?
Given a mixed (some edges directed, some undirected) graph, assign a direction to all the undirected edges to make the graph contain a cycle.
It doesn't "feel" like an NP-hard problem, but I can't think of an easy poly-time algorithm.
What about to make the graph acyclic?
As R B points out in the comments, to make the graph acyclic, you can just use a topological sort on the directed edges (ignoring the undirected edges temporarily) to get an ordering of the vertices, then use that ordering to direct the undirected edges. Therefore, that problem is easy and can be solved in $O(|V|+|E|)$ time.
However, orienting the edges to create a cycle is more tricky. Consider the graph:
A / \ B --> C
In this case, simply sorting the graph to get A B C and then pointing the AC and AB edges backwards does not reveal the cycle.