Improved (In-)Approximability Bounds for d-Scattered Set

Katsikarelis, Ioannis; Lampis, Michael; Paschos, Vangélis Th.

doi:10.1007/978-3-030-39479-0_14

“…The idea of designing super-polynomial time approximation algorithms which obtain guarantees better than those possible in polynomial time has attracted much attention in the last decade [4,10,16,18,20,22,31]. As mentioned, the result closest to the timeapproximation trade-off we give in this paper is the approximation algorithm for Max Min VC given by Bonnet et al [8].…”

Section: Related Workmentioning

confidence: 88%

(In)approximability of maximum minimal FVS

Dublois

¹

,

Hanaka

²

,

Ghadikolaei

³

et al. 2022

Journal of Computer and System Sciences

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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.

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“…Given such a graph, the same preprocessing of Section 3 could be used to derive an analogue of Lemma 11, i.e., a graph G ′ of treewidth O(polylog(n/ε)) could be computed for the net N . Moreover, a 2-approximation for k-Center can be computed in polynomial time on any graph [HS86], and if the input has treewidth t a (1 + ε)-approximation can be computed in (t/ε) O(t) n O(1) time [KLP17]. Using the same arguments to prove Theorem 2 for STP and TSP, it would now be possible to compute a (1 + ε)-approximation for k-Center in quasi-polynomial time (cf.…”

Section: Discussionmentioning

confidence: 99%

Travelling on Graphs with Small Highway Dimension

Disser

¹

,

Feldmann

²

,

Klimm

³

et al. 2021

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show abstract

“…The idea of designing super-polynomial time approximation algorithms which obtain guarantees better than those possible in polynomial time has attracted much attention in the last decade [4,10,16,18,20,22,34]. As mentioned, the result closest to the timeapproximation trade-off we give in this paper is the approximation algorithm for Max Min VC given by Bonnet et al [8].…”

Section: Related Workmentioning

confidence: 97%

(In)approximability of Maximum Minimal FVS

Dublois¹,

Hanaka²,

Ghadikolaei³

et al. 2020

Preprint

Self Cite

0

View full text Add to dashboard Cite

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

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Improved (In-)Approximability Bounds for d-Scattered Set

Cited by 5 publications

References 29 publications

(In)approximability of maximum minimal FVS

(In)approximability of maximum minimal FVS

Travelling on Graphs with Small Highway Dimension

(In)approximability of Maximum Minimal FVS

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