Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

Dublois, Louis; Lampis, Michael; Paschos, Vangélis Th.

doi:10.48550/arxiv.2101.07550

Cited by 2 publications

(3 citation statements)

References 23 publications

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“…We improve this lower bound by showing in Theorem 7 that Up-Dom cannot be solved in time f (k) • n o (k) for any computable function f , implying the same result (replacing k by β) for MMHS. We point out that very recently and independently from our work, this improved lower bound for Up-Dom has also been proved by Dublois et al [17].…”

Section: Contribution and Related Worksupporting

confidence: 82%

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Parameterized complexity of computing maximum minimal blocking and hitting sets

Araújo¹,

Bougeret²,

Campos³

et al. 2021

Preprint

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A blocking set in a graph G is a subset of vertices that intersects every maximum independent set of G. Let mmbs(G) be the size of a maximum (inclusion-wise) minimal blocking set of G. This parameter has recently played an important role in the kernelization of Vertex Cover parameterized by the distance to a graph class F. Indeed, it turns out that the existence of a polynomial kernel for this problem is closely related to the property that mmbs(F) = sup G∈F mmbs(G) is bounded by a constant, and thus several recent results focused on determining mmbs(F) for different classes F. We consider the parameterized complexity of computing mmbs under various parameterizations, such as the size of a maximum independent set of the input graph and the natural parameter. We provide a panorama of the complexity of computing both mmbs and mmhs, which is the size of a maximum minimal hitting set of a hypergraph, a closely related parameter. Finally, we consider the problem of computing mmbs parameterized by treewidth, especially relevant in the context of kernelization. Given the "counting" nature of mmbs, it does not seem to be expressible in monadic second-order logic, hence its tractability does not follow from Courcelle's theorem. Our main technical contribution is a fixed-parameter tractable algorithm for this problem. ACM Subject ClassificationMathematics of computing → Graph algorithms.

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Section: Contribution and Related Worksupporting

confidence: 82%

“…for any computable function f : N → N. We also mention that very recently and independently from our work, Theorem 7 has also been proved by Dublois et al [17], by using a reduction quite similar to ours.…”

Section: Theorem 5 B(b(a)) = Asupporting

confidence: 71%

Parameterized complexity of computing maximum minimal blocking and hitting sets

Araújo¹,

Bougeret²,

Campos³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…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

Self Cite

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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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Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

Cited by 2 publications

References 23 publications

Parameterized complexity of computing maximum minimal blocking and hitting sets

Parameterized complexity of computing maximum minimal blocking and hitting sets

Minimum Stable Cut and Treewidth

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