W[1]-hard problems with FPT time approximation algorithms
In the Directed Odd Cycle Transversal problem the input is a graph $G$ and the task is to find a smallest set $S$ of vertices such that $G-S$ has no (directed) cycles of odd length. In the parameterized version we are also given an integer $k$ and asked whether a solution of size at most $k$ exists. 

In this paper we prove that (R1) the problem is W$[1]$-hard, (R2) that it admits a factor 2 approximation algorithm in time $2^{k^{O(1)}}n^{O(1)}$, and that (R3’) assuming a hypothesis somewhat stronger than $FPT \neq W[1]$ that there exists an $\epsilon &gt; 0$ such that the problem does not admit a $(1+\epsilon)$ approximation algorithm in time $f(k)n^{O(1)}$.