Maximum Minimal Vertex Cover Parameterized by Vertex Cover

Zehavi, Meirav

doi:10.1137/16m109017x

Cited by 18 publications

(8 citation statements)

References 22 publications

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“…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]. 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].…”

Section: Related Workmentioning

confidence: 99%

(In)approximability of maximum minimal FVS

Dublois

Hanaka

Ghadikolaei

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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Section: Related Workmentioning

confidence: 99%

(In)approximability of maximum minimal FVS

Dublois

Hanaka

Ghadikolaei

et al. 2022

Journal of Computer and System Sciences

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“…The motivation behind this line of research is to provide bounds and a refined analysis of such basic heuristics. Problems that have been considered under this lens are Max Min Dominating Set [8,25], Max Min Vertex Cover [16,56],Max Min Separator [40], Max Min Cut [29], Min Max Knapsack [4,34,38], Max Min Edge Cover [47], Max Min Feedback Vertex Set [24]. 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 [43]) and Grundy Coloring, which can be seen as a Max Min version of Coloring [1,9].…”

Section: Related Workmentioning

confidence: 99%

Minimum Stable Cut and Treewidth

Lampis¹

2021

Preprint

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A stable or locally-optimal cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertex. Equivalently, a cut is stable if all vertices have the (weighted) majority of their neighbors on the other side. Finding a stable cut is a prototypical PLS-complete problem that has been studied in the context of local search and of algorithmic game theory.In this paper we study Min Stable Cut, the problem of finding a stable cut of minimum weight, which is closely related to the Price of Anarchy of the Max Cut game. Since this problem is NP-hard, we study its complexity on graphs of low treewidth, low degree, or both. We begin by showing that the problem remains weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this hardness with a pseudo-polynomial DP algorithm solving the problem in time (∆ • W ) O(tw) n O(1) , where tw is the treewidth, ∆ the maximum degree, and W the maximum weight. On the other hand, bounding ∆ is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Min Stable Cut by both tw and ∆ and obtain an FPT algorithm running in time 2 O(∆tw) (n + log W ) O(1) . Our main result for the weighted problem is to provide a reduction showing that both aforementioned algorithms are essentially optimal, even if we replace treewidth by pathwidth: if there exists an algorithm running in (nW ) o(pw) or 2 o(∆pw) (n + log W ) O(1) , then the ETH is false. Complementing this, we show that we can, however, obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions, that is, solutions where no single vertex can unilaterally increase the weight of its incident cut edges by more than a factor of (1 + ε).Motivated by these mostly negative results, we consider Unweighted Min Stable Cut. Here our results already imply a much faster exact algorithm running in time ∆ O(tw) n O(1) . We show that this is also probably essentially optimal: an algorithm running in n o(pw) would contradict the ETH. ACM Subject ClassificationMathematics of computing→Graph algorithms; Theory of Computation → Design and Analysis of Algorithms → Parameterized Complexity and Exact Algorithms

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“…We can also define minimal cuts for disconnected graphs (See Section 2). Maximum Minimal Cut is the following problem: given a graph G = (V, E) and an integer k, determine the existence of a minimal cut (S, V \ S) of size at least k. This type of problems, finding a maximum minimal (or minimum maximal) solution on graphs such as Maximum Minimal Vertex Cover [8,48], Maximum Minimal Dominating Set [1], Maximum Minimal Edge Cover [36], Maximum Minimal Separator [33],…”

Section: Introductionmentioning

confidence: 99%

Parameterized Algorithms for Maximum Cut with Connectivity Constraints

Eto,

Hanaka,

Kobayashi

et al. 2019

Preprint

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We study two variants of Maximum Cut, which we call Connected Maximum Cut and Maximum Minimal Cut, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NPcomplete even on planar graphs whereas Maximum Cut on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-width, and twin-cover number. Finally, we obtain FPT algorithms with respect to the solution size. ACM Subject Classification Mathematics of computing → Graph algorithmsKeywords and phrases Maximum cut, Parameterized algorithm, NP-hardness, Graph parameter Acknowledgements We thank Akitoshi Kawamura and Yukiko Yamauchi for giving an opportunity to discuss in the Open Problem Seminar in Kyushu University.

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Maximum Minimal Vertex Cover Parameterized by Vertex Cover

Cited by 18 publications

References 22 publications

(In)approximability of maximum minimal FVS

(In)approximability of maximum minimal FVS

Minimum Stable Cut and Treewidth

Parameterized Algorithms for Maximum Cut with Connectivity Constraints

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