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 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.

Highlights

  • In a graph G = (V, E), a set S ⊆ V is called a feedback vertex set if the subgraph induced by V \ S is a forest

  • 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: we show that under the Exponential Time Hypothesis (ETH), for any ratio r and > 0, no algorithm can r-approximate this problem in time nO((n/r3/2)1− ), 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

  • This type of time-approximation trade-off was extensively studied by Bonnet et al [8], who showed that Max Min Vertex Cover admits an r-approximation in time 2O(n/r2) and this is optimal under the randomized ETH

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Summary

Introduction

Having settled the approximability of Max Min FVS in polynomial time, we consider the question of how much time needs to be invested if one wishes to guarantee an approximation ratio of r (which may depend on n) where r < n2/3 This type of time-approximation trade-off was extensively studied by Bonnet et al [8], who showed that Max Min Vertex Cover admits an r-approximation in time 2O(n/r2) and this is optimal under the randomized ETH. As explained in [8], current lower bound techniques can rule out improvements in the running time that shave at least n from the exponent, but not improvements which shave poly-logarithmic factors, due to the state of the art in quasi-linear PCP constructions Such improvements are sometimes possible [4] and are conceivable for Max Min VC and Max Min FVS. Lower bounds for this type of algorithm rely on the (randomized) Exponential Time Hypothesis (ETH), which states that there is no (randomized) algorithm for 3-SAT running in time 2o(n)

Preliminaries
Polynomial Time Approximation Algorithm
Basic Reduction Rules and Combinatorial Tools
Polynomial Time Approximation and Extremal Results
Sub-exponential Time Approximation
Hardness of Approximation in Polynomial Time
Hardness of Approximation in Sub-Exponential Time
Conclusions
Full Text
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