On the Complexity of the Upper r-Tolerant Edge Cover Problem

Harutyunyan, Ararat; Ghadikolaei, Mehdi Khosravian; Melissinos, Nikolaos; Monnot, Jérôme; Pagourtzis, Aris

doi:10.1007/978-3-030-57852-7_3

“…On the other hand, UDS and Max Min VC are well-studied problems, both in the context of approximation and in the context of parameterized complexity [1,5,9,11,13,14,19,28,30,34,36]. Many other classical optimization problems have recently been studied in the MaxMin or MinMax framework, such as Max Min Separator [25], Max Min Cut [21], Min Max Knapsack (also known as the Lazy Bureaucrat Problem) [3,23,24], and Max Min Edge Cover [32,26]. Some problems in this area also arise naturally in other forms and have been extensively studied, such as Min Max Matching (also known as Edge Dominating Set [29]) and Grundy Coloring, which can be seen as a Max Min version of Coloring [2,6].…”

Section: Related Workmentioning

confidence: 99%

(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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“…On the other hand, UDS and Max Min VC are well-studied problems, both in the context of approximation and in the context of parameterized complexity [1,5,9,11,13,14,19,28,33,38,40]. Many other classical optimization problems have recently been studied in the MaxMin or MinMax framework, such as Max Min Separator [25], Max Min Cut [21], Min Max Knapsack (also known as the Lazy Bureaucrat Problem) [3,23,24], and some variants of Max Min Edge Cover [35,26]. Some problems in this area also arise naturally in other forms and have been extensively studied, such as Min Max Matching (also known as Edge Dominating Set [32]), Grundy Coloring, which can be seen as a Max Min version of Coloring [2,6], and Max Min VC in hypergraphs, which is known as Upper Transversal [37,29,30,31].…”

Section: Related Workmentioning

confidence: 99%

(In)approximability of Maximum Minimal FVS

Dublois¹,

Hanaka²,

Ghadikolaei³

et al. 2020

Preprint

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0

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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. ACM Subject ClassificationMathematics of computing→Graph algorithms; Theory of Computation → Design and Analysis of Algorithms → Approximation algorithms analysis

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“…A remark on the collaboration.. This work was initiated by Jérôme, who had a long-standing interest in the algorithmic complexity of min-max / max-min variants of graph parameters, and had published many nice papers on this topic [1,2,7,8,3,9,28,22,29]. He kindly invited the second and third authors to join him in a new collaboration on upper edge domination; some of Jérôme's papers on "upper" graph parameters were co-authored with the second author [7,8,9], and the third author's PhD thesis also dealt with this topic [32].…”

Section: Introductionmentioning

confidence: 99%