Best algorithm for detecting cycles in a directed graph

Given that this is a schedule of jobs, I suspect that at some point you are going to sort them into a proposed order of execution.

If that's the case, then a topological sort implementation may in any case detect cycles. UNIX tsort certainly does. I think it is likely that it is therefore more efficient to detect cycles at the same time as tsorting, rather than in a separate step.

So the question might become, "how do I most efficiently tsort", rather than "how do I most efficiently detect loops". To which the answer is probably "use a library", but failing that the following Wikipedia article:

http://en.wikipedia.org/wiki/Topological_sorting

has the pseudo-code for one algorithm, and a brief description of another from Tarjan. Both have O(|V| + |E|) time complexity.

Tarjan's strongly connected components algorithm has O(|E| + |V|) time complexity.

For other algorithms, see Strongly connected components on Wikipedia.