On the maximum weight minimal separator

Hanaka, Tesshu; Bodlaender, Hans L.; Zanden, Tom C. van der; Ono, Hiroyuki

doi:10.1016/j.tcs.2019.09.025

Cited by 12 publications

(15 citation statements)

References 22 publications

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“…In this paper, we strengthen their hardness result and give an even faster correct algorithm for Maximum Minimal Separator than the previous claimed algorithm in [22].…”

Section: Our Contributionsupporting

confidence: 61%

“…Motivated by these results, Hanaka et al [22] studied the problem of finding a maximum cardinality minimal separator of graphs. Formally, the problem is defined as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Formally, the problem is defined as follows. bipartite [22] bipartite permutation permutation comparability d-trapezoid cocomparability of bounded dimension [4] co-comparability AT-free chordal interval circular-arc [24] co-bipartite [Thm. 2] weakly chordal [7] proper interval claw-free line [Thm.…”

Section: Introductionmentioning

confidence: 99%

“…2] weakly chordal [7] proper interval claw-free line [Thm. 3] circle [24] distance hereditary restricted to unweighted bipartite graphs [22]. They also proved that the problem can be solved in polynomial time for bounded treewidth graphs.…”

Section: Introductionmentioning

confidence: 99%

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Finding a Maximum Minimal Separator: Graph Classes and Fixed-Parameter Tractability

Hanaka¹,

Kobayashi²,

Kobayashi³

et al. 2020

Preprint

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We study the problem of finding a maximum cardinality minimal separator of a graph. This problem is known to be NP-hard even for bipartite graphs. In this paper, we strengthen this hardness by showing that for planar bipartite graphs, the problem remains NP-hard. Moreover, for co-bipartite graphs and for line graphs, the problem also remains NP-hard. On the positive side, we give an algorithm deciding whether an input graph has a minimal separator of size at least k that runs in time 2 O(k) n O(1) . We further show that a subexponential parameterized algorithm does not exist unless the Exponential Time Hypothesis (ETH) fails. Finally, we discuss a lower bound for polynomial kernelizations of this problem.

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“…In this paper, we strengthen their hardness result and give an even faster correct algorithm for Maximum Minimal Separator than the previous claimed algorithm in [22].…”

Section: Our Contributionsupporting

confidence: 61%

“…Motivated by these results, Hanaka et al [22] studied the problem of finding a maximum cardinality minimal separator of graphs. Formally, the problem is defined as follows.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Finding a Maximum Minimal Separator: Graph Classes and Fixed-Parameter Tractability

Hanaka¹,

Kobayashi²,

Kobayashi³

et al. 2020

Preprint

Self Cite

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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 [1,5,9,11,13,14,19,28,30,34,36]. 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

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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