Parameterized Max Min Feedback Vertex Set
Michael Lampis, Nikolaos Melissinos, Manolis Vasilakis
TL;DR
This work investigates the parameterized complexity of Max Min FVS, where one seeks a maximal minimal feedback vertex set of size at least $k$. It delivers a treewidth-based DP running in $\text{tw}^{O(\text{tw})} n^{O(1)}$ and proves this is essentially optimal under ETH, by showing a $\text{vc}^{o(\text{vc})} n^{O(1)}$ lower bound. The authors also provide a corrected single-exponential $9.34^k n^{O(1)}$ branching algorithm, improving the previously claimed $10^k n^{O(1)}$. Together, these results delineate the limits of single-exponential algorithms for treewidth and vertex-cover parameterizations of Max Min FVS and contribute to a deeper understanding of when Cut&Count techniques apply to feedback-vertex-set variants.
Abstract
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 $9.34^k n^{O(1)}$.