Any fast algorithm for minimum cost feedback arc set problem?
Daniel Apon linked to the conference version of my paper. I suggest the draft journal version instead: http://www.cs.brown.edu/people/ws/papers/fast_journal.pdf .
On tournament graphs some experimental work suggests that local search does quite well. See Anke van Zuylen and Frans Schalekampf's recent ALENEX paper: http://www.siam.org/proceedings/alenex/2009/alx09_004_schalekampf.pdf .
If the weights satisfy either "probability constraints" or "triangle inequalities" there's a constant-factor approximation algorithm based on quicksort. See Ailon, Charikar and Newman's recent JACM paper.
Can you tell us a bit more about what sorts of instances you have in mind and whether you're looking for something that works well in practice or in theory?