Finding even cycle in directed graphs
Yes, a polynomial time algorithm was first given in:

Neil Robertson, P. D. Seymour, Robin Thomas. "Permanents, Pfaffian orientations, and even directed circuits." Annals of Mathematics 150.3 (1999): 929-975. arXiv

edit: Actually, according to the acknowledgements section of the above paper, the result was first obtained by McCuaig who later published it as:

William McCuaig. "Pólya's Permanent Problem." Electr. J. Comb. 11(1) (2004) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1r79