  The Feedback Vertex Set problem is undoubtedly one of the most well-studied
problems in Parameterized Complexity. In this problem, given an undirected
graph $G$ and a non-negative integer $k$, the objective is to test whether
there exists a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is
a forest. After a long line of improvement, recently, Li and Nederlof [SODA,
2020] designed a randomized algorithm for the problem running in time
$\mathcal{O}^{\star}(2.7^k)$. In the Parameterized Complexity literature,
several problems around Feedback Vertex Set have been studied. Some of these
include Independent Feedback Vertex Set (where the set $S$ should be an
independent set in $G$), Almost Forest Deletion and Pseudoforest Deletion. In
Pseudoforest Deletion, each connected component in $G-S$ has at most one cycle
in it. However, in Almost Forest Deletion, the input is a graph $G$ and
non-negative integers $k,\ell \in \mathbb{N}$, and the objective is to test
whether there exists a vertex subset $S$ of size at most $k$, such that $G-S$
is $\ell$ edges away from a forest. In this paper, using the methodology of Li
and Nederlof [SODA, 2020], we obtain the current fastest algorithms for all
these problems. In particular we obtain following randomized algorithms.
  1) Independent Feedback Vertex Set can be solved in time
$\mathcal{O}^{\star}(2.7^k)$.
  2) Pseudo Forest Deletion can be solved in time
$\mathcal{O}^{\star}(2.85^k)$.
  3) Almost Forest Deletion can be solved in $\mathcal{O}^{\star}(\min\{2.85^k
\cdot 8.54^\ell,2.7^k \cdot 36.61^\ell,3^k \cdot 1.78^\ell\})$.
