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$k$，最大最小反馈顶点集问题询问是否存在一个大小至少为$k$的最小顶点集，删除该集合可破坏所有环。我们针对该问题的结构参数和自然参数，改进了其参数化复杂度的现有成果。首先，利用标准动态规划技术，我们提出一个时间复杂度为$\textrm{tw}^{O(\textrm{tw})}n^{O(1)}$的算法，显著推广了Gaikwad等人近期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$\textrm{vc}^{o(\textrm{vc})}n^{O(1)}$算法都将否定指数时间假设。针对自然参数$k$，前述Gaikwad等人的近期工作声称存在一个复杂性为$10^k n^{O(1)}$的固定参数可解分支算法。我们指出该算法不正确，并给出一个复杂性为$9.34^k n^{O(1)}$的分支算法。