There are several 2-approximation algorithms for the UNWEIGHTED feedback vertex set problem (FVS), which are summarized in . Note that the reduction from vertex cover to FVS is approximation-preserving. Assuming Unique Game Conjecture, we cannot expect better algorithms. The question is:
Is there an unweighted graph on which some of the algorithm really reaches ratio 2?
 contains such a tight instance for weighted FVS.
- Vineet Bafna and Piotr Berman and Toshihiro Fujito, http://doi.org/10.1137/S0895480196305124;
- Ann Becker and Dan Geiger, http://doi.org/10.1016/0004-3702(95)00004-6;
- Toshihiro Fujito, http://doi.org/10.1016/0020-0190(96)00094-4;
- Fabián A. Chudak, Michel X. Goemans, Dorit S. Hochbaum, David P. Williamson, http://doi.org/10.1016/S0167-6377(98)00021-2.