A Simplified Parameterized Algorithm for Directed Feedback Vertex Set

The Directed Feedback Vertex Set problem (DFVS) asks whether it is possible to remove at most $k$ vertices from a directed graph to make it acyclic. Whether DFVS is fixed-parameter tractable was a long-standing open problem in parameterized complexity until it was solved by Chen et al. in 2008 (STOC 2008). Now the running-time bound of this problem is improved to $\mathcal O(k!4^kk^5(n+m))$ (Lokshtanov et al, SODA 2018), where $n$ and $m$ are the numbers of vertices and arcs in the graph. In this paper, we simplify one crucial step in all previous parameterized algorithms for DFVS, which is to solve the compression version of the problem, and refine the running-time bound for DFVS to $\mathcal O(k!2^{o(k)}(n+m))$.