On the Hardness of Approximating Some NP-optimization Problems Related to Minimum Linear Ordering Problem

Mishra, Sambit Kumar; Sikdar, Kripasindhu

doi:10.1051/ita:2001121

“…Intuitively, one may be tempted to think that this problem should be harder than Max Min VC, since hitting cycles is more complex than hitting edges, but easier than UDS, since hitting cycles still offers us more structure than an arbitrary hypergraph. However, to the best of our knowledge, no n 1− -approximation algorithm is currently known for Max Min FVS (so the problem could be as hard as UDS), and the best hardness of approximation bound known is n 1/2− [33] (so the problem could be as easy as Max Min VC).…”

Section: :2 (In)approximability Of Maximum Minimal Fvsmentioning

confidence: 99%

“…This completely settles the approximability of the problem in polynomial time. Along the way, we also give an approximation algorithm with ratio O(∆), show that no algorithm can achieve ratio ∆ 1− , for any > 0, and improve the best known NP-completeness proof for Max Min FVS from ∆ ≥ 9 [33] to ∆ ≥ 6, where ∆ is the maximum degree of the input graph.…”

Section: :2 (In)approximability Of Maximum Minimal Fvsmentioning

confidence: 99%

“…To the best of our knowledge, Max Min FVS was first considered by Mishra and Sikdar [33], who showed that the problem does not admit an n 1/2− approximation (unless P = NP), and that it remains APX-hard for ∆ ≥ 9. 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].…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

(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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“…Considering Max Ext VC on G = (V, E) and the particular subset U = V (resp., Min Ext IS with U = ∅), we obtain two well known optimization problems called upper vertex cover (UVC for short, also called the maximum minimal vertex cover problem) and the maximum minimal independent set problem (equivalently ISDS for short). In [26], the computational complexity of these two problems have been studied (among 12 problems), and (in)approximability results are given in [27,9] for UVC and in [18] for ISDS where lower bounds of O(n ε−1/2 ) and O(n 1−ε ), respectively, for graphs on n vertices are given for every ε > 0. Analogous bounds can be also derived depending on the maximum degree ∆ of the graph.…”

Section: Min Ext Ismentioning

confidence: 99%

Extension of Vertex Cover and Independent Set in Some Classes of Graphs

Casel¹,

Fernau²,

Ghadikoalei³

et al. 2019

Lecture Notes in Computer Science

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We consider extension variants of the classical graph problems Vertex Cover and Independent Set. Given a graph G = (V, E) and a vertex set U ⊆ V , it is asked if there exists a minimal vertex cover (resp. maximal independent set) S with U ⊆ S (resp. U ⊇ S). Possibly contradicting intuition, these problems tend to be NP-hard, even in graph classes where the classical problem can be solved in polynomial time. Yet, we exhibit some graph classes where the extension variant remains polynomial-time solvable. We also study the parameterized complexity of theses problems, with parameter |U |, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All these complexity considerations are also carried out in very restricted scenarios, be it degree or topological restrictions (bipartite, planar or chordal graphs). This also motivates presenting some explicit branching algorithms for degree-bounded instances.We further discuss the price of extension, measuring the distance of U to the closest set that can be extended, which results in natural optimization problems related to extension problems for which we discuss polynomial-time approximability.

show abstract

“…On an intuitive level, one may be tempted to think that this problem should be harder than Max Min VC, since hitting cycles is more complex than hitting edges, but easier than UDS, since hitting cycles still offers us more structure than an arbitrary hypergraph. However, to the best of our knowledge, no n 1− -approximation algorithm is currently known for Max Min FVS (so the problem could be as hard as UDS), and the best hardness of approximation bound known is n 1/2− [36] (so the problem could be as easy as Max Min VC).…”

Section: Introductionmentioning

confidence: 99%

(In)approximability of Maximum Minimal FVS

Dublois¹,

Hanaka²,

Ghadikolaei³

et al. 2020

Preprint

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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On the Hardness of Approximating Some NP-optimization Problems Related to Minimum Linear Ordering Problem

Cited by 7 publications

References 34 publications

(In)approximability of maximum minimal FVS

(In)approximability of maximum minimal FVS

Extension of Vertex Cover and Independent Set in Some Classes of Graphs

(In)approximability of Maximum Minimal FVS

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