Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling

Bläsius, Thomas; Friedrich, Tobias; Lischeid, Julius; Meeks, Kitty; Schirneck, Martin

doi:10.1137/1.9781611975499.11

Cited by 14 publications

(12 citation statements)

References 35 publications

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“…Indeed, the particular case α = 2 of MMHS corresponds to the Maximum Minimal Vertex Cover (MMVC for short) problem, which is NP-hard [7]. When parameterizing by β only, we show in Proposition 6 that MMBS is para-NP-hard, whereas MMHS is W [1]-hard [2] and XP [4]. As discussed in Section 3, the W [1]-hardness proof of [2] implies that, unless the Exponential Time Hypothesis (ETH for short) fails, Up-Dom cannot be solved in time f (k) • n o( √ k) on n-vertex graphs for any computable function f .…”

Section: Contribution and Related Workmentioning

confidence: 99%

“…Finally, when parameterizing by α + β, the hardness result given in Proposition 6 (i.e., parameterizing by α for fixed β) implies that MMBS is W [1]-hard, whereas MMHS is FPT for the following reasons. We first provide in Corollary 18 a simple FPT algorithm for MMHS that reduces to an extension problem considered by Bläsius et al [4], and then design in Theorem 24 a more involved an ad-hoc algorithm to improve the running time to O * (2 αβ ), where the O * -notation hides multiplicative polynomial terms (see Section 2).…”

Section: Contribution and Related Workmentioning

confidence: 99%

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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 Workmentioning

confidence: 99%

Section: Contribution and Related Workmentioning

confidence: 99%

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

“…In fact, enumeration algorithms are often analyzed not only with respect to their running time, but also in terms of space consumption, see [9,14]. For data profiling problems like Enumerate Minimal UCCs on the other hand, space-efficient algorithms have only recently started to received some attention [5,6,36].…”

Section: Approximation and Discoverymentioning

confidence: 99%

“…We show here that the detection of inclusion dependencies has this property, making it the second such problem. Since this result was first announced, Bläsius et al [6] have also proven the extension problem for minimal hitting sets to be W[3]-complete using different techniques. The latter has subsequently been improved by Casel et al [11], they have shown that already the special case of extension to minimal dominating sets in bipartite graphs is hard for W [3].…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Dependency Detection and Discovery in Relational Databases

Bläsius¹,

Friedrich²,

Schirneck³

2021

Preprint

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Multi-column dependencies in relational databases come associated with two different computational tasks. The detection problem is to decide whether a dependency of a certain type and size holds in a given database, the discovery problem asks to enumerate all valid dependencies of that type. We settle the complexity of both of these problems for unique column combinations (UCCs), functional dependencies (FDs), and inclusion dependencies (INDs).We show that the detection of UCCs and FDs is W[2]-complete when parameterized by the solution size. The discovery of inclusion-wise minimal UCCs is proven to be equivalent under parsimonious reductions to the transversal hypergraph problem of enumerating the minimal hitting sets of a hypergraph. The discovery of FDs is equivalent to the simultaneous enumeration of the hitting sets of multiple input hypergraphs.We further identify the detection of INDs as one of the first natural W[3]-complete problems. The discovery of maximal INDs is shown to be equivalent to enumerating the maximal satisfying assignments of antimonotone, 3-normalized Boolean formulas.

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“…The following fundamental combinatorial optimization problem arises in bioinformatics [43], medicine [41,51], clustering [12,36], automatic reasoning [16,25,48], feature selection [15,30], radio frequency allocation [49], software engineering [47], and public transport optimization [14,52].…”

Section: Introductionmentioning

confidence: 99%

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

Bevern¹,

Kirilin²,

Skachkov³

et al. 2021

Preprint

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The NP-hard Multiple Hitting Set problem is finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasibility of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Date) architectures.

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Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling

Cited by 14 publications

References 35 publications

Parameterized complexity of computing maximum minimal blocking and hitting sets

Parameterized complexity of computing maximum minimal blocking and hitting sets

The Complexity of Dependency Detection and Discovery in Relational Databases

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

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