(In)approximability of maximum minimal FVS

We study the approximability of the NP-complete Maximum Minimal Feedback Vertex Set problem. Informally, this natural problem seems to lie in an intermediate space between two more well-studied problems of this type: Maximum Minimal Vertex Cover, for which the best achievable approximation ratio is √n, and Upper Dominating Set, which does not admit any n¹−∊ approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for Max Min FVS with a ratio of O(n^2/3), as well as a matching hardness of approximation bound of n^2/3−∊, improving the previous known hardness of n^1/2−∊. Along the way, we also obtain an O(∆)-approximation and show that this is asymptotically best possible, and we improve the bound for which the problem is NP-hard from ∆ ≥ 9 to ∆ ≥ 6. Having settled the problem’s approximability in polynomial time, we move to the context of super-polynomial time. We devise a generalization of our approximation algorithm which, for any desired approximation ratio r, produces an r-approximate solution in time n^O(n/r3/2). This time-approximation trade-off is essentially tight: we show that under the ETH, for any ratio r and ∊ > 0, no algorithm can r-approximate this problem in time n^{O((n/r3/2)1−∊)}, hence we precisely characterize the approximability of the problem for the whole spectrum between polynomial and sub-exponential time, up to an arbitrarily small constant in the second exponent.