Computing metric hulls in graphs

Kolja Knauer 1 Nicolas Nisse 2
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : 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 confirms a conjecture of Albenque and Knauer and 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 computing if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-complete if only the ground set is given. A special instance of this problem is computing the dimension of a poset given its linear extension graph, that was 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 Σ P 2 complete.
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[Research Report] Inria - Sophia Antipolis. 2017
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Soumis le : vendredi 6 octobre 2017 - 22:41:28
Dernière modification le : mardi 10 octobre 2017 - 13:47:22

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Kolja Knauer, Nicolas Nisse. Computing metric hulls in graphs. [Research Report] Inria - Sophia Antipolis. 2017. 〈hal-01612515〉

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