Simultaneous Feedback Vertex Set: A Parameterized Perspective

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).