W[1]-hard problems with FPT time approximation algorithms I'm looking for problems that are hard to solve in FPT time but has an approximation algorithm. That is, problems that are:

R1. W[1]-hard.

R2. Admit a (preferably constant) approximation algorithm in FPT time.

The problem I'm familiar with is counting the number of simple paths of length $k$ in a graph. It is known to be #W[1]-hard, but admits a $(1+\epsilon)$-approximation in FPT time (for any constant $\epsilon$).

Also interesting would be problems that satisfy R1 and R2, and also:

R3. There exists $\epsilon&gt;0$ such that the problem is not $(1+\epsilon)$ approximable in FPT time (unless W[1]=FPT).


  What other problems satisfy R1 and R2, and possibly R3?
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 $f(k)n^{O(1)}$.