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−ϵ approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for MAXIMUM MINIMAL FEEDBACK VERTEX SET with a ratio of O(n2/3), as well as a matching hardness of approximation bound of n2/3−ϵ, improving the previously known hardness of n1/2−ϵ. 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 nO(n/r3/2). This time-approximation trade-off is essentially tight under the ETH.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computer Networks and Communications
- Computational Theory and Mathematics
- Applied Mathematics