(In)approximability of Maximum Minimal FVS

Abstract

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 n 1− 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/r 3/2 ) . 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/r 3/2 ) 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. ACM Subject ClassificationMathematics of computing→Graph algorithms; Theory of Computation → Design and Analysis of Algorithms → Approximation algorithms analysis