Given a graph $G$ and an integer $k$, Max Min FVS asks whether there exists a minimal set of vertices of size at least $k$ whose deletion destroys all cycles. We present several results that improve upon the state of the art of the parameterized complexity of this problem with respect to both structural and natural parameters. Using standard DP techniques, we first present an algorithm of time $\textrm{tw}^{O(\textrm{tw})}n^{O(1)}$, significantly generalizing a recent algorithm of Gaikwad et al. of time $\textrm{vc}^{O(\textrm{vc})}n^{O(1)}$, where $\textrm{tw}, \textrm{vc}$ denote the input graph's treewidth and vertex cover respectively. Subsequently, we show that both of these algorithms are essentially optimal, since a $\textrm{vc}^{o(\textrm{vc})}n^{O(1)}$ algorithm would refute the ETH. With respect to the natural parameter $k$, the aforementioned recent work by Gaikwad et al. claimed an FPT branching algorithm with complexity $10^k n^{O(1)}$. We point out that this algorithm is incorrect and present a branching algorithm of complexity $9.34^k n^{O(1)}$.

翻译：给定图$G$和整数$k$，Max Min FVS问题询问是否存在大小至少为$k$的最小顶点集，删除该集合中的所有顶点可消除所有环。我们提出了若干结果，在结构参数和自然参数两方面改进了该问题参数化复杂度的现有研究水平。首先，利用标准DP技术，我们提出时间复杂度为$\textrm{tw}^{O(\textrm{tw})}n^{O(1)}$的算法，显著推广了Gaikwad等人最近提出的时间复杂度为$\textrm{vc}^{O(\textrm{vc})}n^{O(1)}$的算法，其中$\textrm{tw}$和$\textrm{vc}$分别表示输入图的树宽和顶点覆盖。随后，我们证明这两个算法本质上是最优的，因为任何$\textrm{vc}^{o(\textrm{vc})}n^{O(1)}$算法都将反驳ETH。关于自然参数$k$，前述Gaikwad等人最近的工作声称存在复杂度为$10^k n^{O(1)}$的FPT分支算法。我们指出该算法存在错误，并提出了一个复杂度为$9.34^k n^{O(1)}$的分支算法。