  Given a family of graphs $\mathcal{F}$, a graph $G$, and a positive integer
$k$, the $\mathcal{F}$-Deletion problem asks whether we can delete at most $k$
vertices from $G$ to obtain a graph in $\mathcal{F}$. $\mathcal{F}$-Deletion
generalizes many classical graph problems such as Vertex Cover, Feedback Vertex
Set, and Odd Cycle Transversal. A graph $G = (V, \cup_{i=1}^{\alpha} E_{i})$,
where the edge set of $G$ is partitioned into $\alpha$ color classes, is called
an $\alpha$-edge-colored graph. A natural extension of the
$\mathcal{F}$-Deletion problem to edge-colored graphs is the
$\alpha$-Simultaneous $\mathcal{F}$-Deletion problem. In the latter problem, we
are given an $\alpha$-edge-colored graph $G$ and the goal is to find a set $S$
of at most $k$ vertices such that each graph $G_i \setminus S$, where $G_i =
(V, E_i)$ and $1 \leq i \leq \alpha$, is in $\mathcal{F}$. In this work, we
study $\alpha$-Simultaneous $\mathcal{F}$-Deletion for $\mathcal{F}$ being the
family of forests. In other words, we focus on the $\alpha$-Simultaneous
Feedback Vertex Set ($\alpha$-SimFVS) problem. Algorithmically, we show that,
like its classical counterpart, $\alpha$-SimFVS parameterized by $k$ is
fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed
constant $\alpha$. In particular, we give an algorithm running in $2^{O(\alpha
k)}n^{O(1)}$ time and a kernel with $O(\alpha k^{3(\alpha + 1)})$ vertices. The
running time of our algorithm implies that $\alpha$-SimFVS is FPT even when
$\alpha \in o(\log n)$. We complement this positive result by showing that for
$\alpha \in O(\log n)$, where $n$ is the number of vertices in the input graph,
$\alpha$-SimFVS becomes W[1]-hard. Our positive results answer one of the open
problems posed by Cai and Ye (MFCS 2014).
