Computing metric hulls in graphs

TL;DR

This paper investigates the computational complexity of calculating metric hulls and related structures in graphs, establishing polynomial-time algorithms for some cases and proving NP-completeness and higher complexity for others.

Contribution

It proves polynomial-time computability of the smallest preimage of closed sets and convex hull-number, and establishes NP-completeness and $ ext{LOGSNP}$-completeness for related problems, introducing the concept of isometric hulls.

Findings

Polynomial-time algorithm for smallest preimage of closed sets.

NP-completeness of computing isometric hulls for three vertices.

$ ext{LOGSNP}$-completeness of determining if the preimage size is logarithmic.

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 $\Sigma^P_2$ complete.