Faster Fixed Parameter Tractable Algorithms for Finding Feedback Vertex Sets

A feedback vertex set (fvs) of a graph is a set of vertices whose removal results in an acyclic graph. We show that if an undirected graph on n vertices with minimum degree at least 3 has a fvs on at most 1/3n(1-epsilon) vertices, then there is a cycle of length at most 6/epsilon (for epsilon >= 1/2, we can even improve this to just 6). Using this, we obtain a O (( 12 logk/log log + 6)(k) n(omega)) algorithm for testing whether an undirected graph on n vertices has a fvs of size at most k. Here n(omega) is the complexity of the best matrix multiplication algorithm. The previous best parameterized algorithm for this problem took O ((2k + 1)(k) n(2)) time. We also investigate the fixed parameter complexity of weighted feedback vertex set problem in weighted undirected graphs.