Computing metric hulls in graphs

We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This implies that there is a polynomial time algorithm to compute the convex hull number of a graph, when all its convex subgraphs are given as input. We t...

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Veröffentlicht in:Discrete Mathematics and Theoretical Computer Science Jg. 21 no. 1, ICGT 2018; H. 1; S. COV1 - 20
Hauptverfasser: Knauer, Kolja, Nisse, Nicolas
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Nancy DMTCS 01.05.2019
Discrete Mathematics & Theoretical Computer Science
Schlagworte:
[info.info-cc]computer science [cs]/computational complexity [cs.cc]
Algorithms
Apexes
Applied research
Completeness
Complexity theory
Computational Complexity
Computational geometry
Computer Science
Convexity
Graph theory
Graphs
Hulls
Polynomials
Set theory
Shortest-path problems
France
ISSN:1365-8050, 1462-7264, 1365-8050
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Zusammenfassung:We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This implies that there is a polynomial time algorithm to compute the convex hull number of a graph, when all its convex subgraphs are given as input. We then show that deciding if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-hard if only the ground set is given. A special instance of this problem is to compute the dimension of a poset given its linear extension graph, that is conjectured to be in P.The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull number of a graph and show that computing it is $\Sigma^P_2$ complete.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1365-8050
1462-7264
1365-8050
DOI:10.23638/DMTCS-21-1-11