On the Complexity of Solution Extension of Optimization Problems

Casel, Katrin; Fernau, Henning; Ghadikolaei, Mehdi Khosravian; Monnot, Jérôme; Sikora, Florian

doi:10.48550/arxiv.1810.04553

Cited by 2 publications

(3 citation statements)

References 24 publications

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“…Proof. For k = 3, the result follows from [5] and the previous discussion. Now, let us prove it for k = 4.…”

Section: P K -Free Graphsmentioning

confidence: 52%

“…It is proven in [5], among other results, that the extension problem with A = ∅ is NP-complete in the case of maximal matchings. Since maximal P 3 -free graphs are exactly maximal matchings in triangle-free graphs, this result implies that the edge extension problem for P 3 -free graphs is NP-complete, even in triangle-free graphs and with A = ∅.…”

Section: P K -Free Graphsmentioning

confidence: 99%

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Efficient enumeration of maximal split subgraphs and induced sub-cographs and related classes

Brosse¹,

Lagoutte²,

Limouzy³

et al. 2020

Preprint

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In this paper, we are interested in algorithms that take in input an arbitrary graph G, and that enumerate in output all the (inclusion-wise) maximal "subgraphs" of G which fulfil a given property Π. All over this paper, we study several different properties Π, and the notion of subgraph under consideration (induced or not) will vary from a result to another.More precisely, we present efficient algorithms to list all maximal split subgraphs, sub-cographs and some subclasses of cographs of a given input graph. All the algorithms presented here run in polynomial delay, and moreover for split graphs it only requires polynomial space. In order to develop an algorithm for maximal split (edge-)subgraphs, we establish a bijection between the maximal split subgraphs and the maximal independent sets of an auxiliary graph. For cographs and some subclasses , the algorithms rely on a framework recently introduced by Conte & Uno [9] called Proximity Search. Finally we consider the extension problem, which consists in deciding if there exists a maximal induced subgraph satisfying a property Π that contains a set of prescribed vertices and that avoids another set of vertices. We show that this problem is NP-complete for every "interesting" hereditary property Π. We extend the hardness result to some specific edge version of the extension problem.

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“…Proof. For k = 3, the result follows from [5] and the previous discussion. Now, let us prove it for k = 4.…”

Section: P K -Free Graphsmentioning

confidence: 52%

Section: P K -Free Graphsmentioning

confidence: 99%

Efficient enumeration of maximal split subgraphs and induced sub-cographs and related classes

Brosse¹,

Lagoutte²,

Limouzy³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…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]. Finally, building on the work presented here, Hannula, Song, and Link [32] have very recently identified independence detection in relational databases as another representative of this class.…”

Section: Introductionmentioning

confidence: 57%

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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On the Complexity of Solution Extension of Optimization Problems

Cited by 2 publications

References 24 publications

Efficient enumeration of maximal split subgraphs and induced sub-cographs and related classes

Efficient enumeration of maximal split subgraphs and induced sub-cographs and related classes

The Complexity of Dependency Detection and Discovery in Relational Databases

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