The many facets of upper domination

Bazgan, Cristina; Branković, Ljiljana; Casel, Katrin; Fernau, Henning; Jansen, Klaus; Klein, Kim-Manuel; Lampis, Michael; Liedloff, Mathieu; Monnot, Jérôme; Paschos, Vangélis Th.

doi:10.1016/j.tcs.2017.05.042

Cited by 38 publications

(38 citation statements)

References 33 publications

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“…At first glance, one would expect Max Min VC to be the easier of these two problems: both problems can be seen as trying to find the largest minimal hitting set of a hypergraph, but in the case of Max Min VC the hypergraph has a very restricted structure, while in UDS the hypergraph is essentially arbitrary. This intuition turns out to be correct: while UDS admits no n 1− -approximation [5], Max Min VC admits a √ n-approximation (but no n 1/2− -approximation) [9].…”

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

confidence: 95%

“…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%

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

confidence: 95%

Section: Related Workmentioning

confidence: 99%

(In)approximability of maximum minimal FVS

Dublois

Hanaka

Ghadikolaei

et al. 2022

Journal of Computer and System Sciences

Self Cite

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“…To show membership in W [3] for Independent Family, we construct a polynomial parameter-preserving reduction to a circuit of bounded depth and weft 3. Selecting some set in system S corresponds to setting an input node to true, the covering constraint is modelled via large gates representing the elements of the union, the sets in T and a final large (negated) OR-gate.…”

Section: Independent Familymentioning

confidence: 99%

“…Note that the rank k * is not known to the algorithms, the input consists only of the hypergraph itself. In fact, it has been shown recently by Bazgan et al that computing k * is W [1]-hard (parameterised by k * ); also, it cannot be approximated with a factor of n 1−ε for any ε > 0, unless P = NP [3]. The upper bound on the delay holds regardless of the knowledge of k * .…”

Section: Solving the Extensionmentioning

confidence: 99%

“…After finding all solutions, Apriori may take some additional time to terminate as it needs to verify that the search space of future candidates is in fact empty. It is generally agreed upon that this is hard to avoid, finding an efficient stopping criterion here would immediately yield faster algorithms to compute k * , see [3] and [28] for a more detailed discussion. To make this transparent in our experiments, we measured two run times for the brute-force algorithm: the enumeration time is the time it took to find all minimal hitting sets, and the completion time is the total run time until termination.…”

Section: Oracle Callsmentioning

confidence: 99%

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Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling

Bläsius

Friedrich

Lischeid

et al. 2019

2019 Proceedings of the Twenty-First Workshop on Algorithm Engineering and Experiments (ALENEX)

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We devise an enumeration method for inclusion-wise minimal hitting sets in hypergraphs. It has delay O(m k * +1 • n 2 ) and uses linear space. Hereby, n is the number of vertices, m the number of hyperedges, and k * the rank of the transversal hypergraph. In particular, on classes of hypergraphs for which the cardinality k * of the largest minimal hitting set is bounded, the delay is polynomial. The algorithm solves the extension problem for minimal hitting sets as a subroutine. We show that the extension problem is W [3]-complete when parameterised by the cardinality of the set which is to be extended. For the subroutine, we give an algorithm that is optimal under the exponential time hypothesis. Despite these lower bounds, we provide empirical evidence showing that the enumeration outperforms the theoretical worst-case guarantee on hypergraphs arising in the profiling of relational databases, namely, in the detection of unique column combinations.

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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.

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The many facets of upper domination

Cited by 38 publications

References 33 publications

(In)approximability of maximum minimal FVS

(In)approximability of maximum minimal FVS

Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling

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

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