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(In)approximability of Maximum Minimal FVS

Authors Louis Dublois, Tesshu Hanaka , Mehdi Khosravian Ghadikolaei, Michael Lampis , Nikolaos Melissinos

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Author Details

Louis Dublois
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, Paris, France
Tesshu Hanaka
  • Chuo University, Tokyo, Japan
Mehdi Khosravian Ghadikolaei
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, Paris, France
Michael Lampis
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, Paris, France
Nikolaos Melissinos
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, Paris, France

Funding

This work is partially supported by PRC CNRS JSPS project PARAGA and by JSPS KAKENHI Grant Number JP19K21537.

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Louis Dublois, Tesshu Hanaka, Mehdi Khosravian Ghadikolaei, Michael Lampis, and Nikolaos Melissinos. (In)approximability of Maximum Minimal FVS. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.3

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-ε} 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-ε}, improving the previous known hardness of n^{1/2-ε}. Along the way, we also obtain an O(Δ)-approximation and show that this is asymptotically best possible, and we improve the bound for which the problem is NP-hard from Δ ≥ 9 to Δ ≥ 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 ε > 0, no algorithm can r-approximate this problem in time n^{O((n/r^{3/2})^{1-ε})}, 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Approximation Algorithms
  • ETH
  • Inapproximability

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