Faster deterministic Feedback Vertex Set

We present two new deterministic algorithms for the Feedback Vertex Set problem parameterized by the solution size. We begin with a simple algorithm, which runs in O((2 + φ)) time, where φ < 1.619 is the golden ratio. It already surpasses the previously fastest O((1+ 2 √ 2))-time deterministic algorithm due to Cao et al. [SWAT 2010]. In our developments we follow the approach of Cao et al., however, thanks to a new reduction rule, we obtain not only better dependency on the parameter in the running time, but also a solution with simple analysis and only a single branching rule. Then, we present a modification of the algorithm which, using a more involved set of branching rules, achieves O(3.592) running time.