The hypermetric cone on $8$ vertices and some generalizations

TL;DR

This paper computes the facets and extreme rays of the hypermetric cone on 8 vertices, introduces algorithms for the hypermetric polytope, and explores various generalizations of hypermetrics.

Contribution

It provides the first detailed enumeration of facets and extreme rays of the hypermetric cone on 8 vertices and offers algorithms for the hypermetric polytope for small dimensions.

Findings

Facets of HYP_8: 298,592 in 86 orbits

Extreme rays of HYP_8: 242,695,427 in 9,003 orbits

Algorithms for hypermetric polytope for n ≤ 8

Abstract

The lists of facets -- $298,592$ in $86$ orbits -- and of extreme rays -- $242,695,427$ in $9,003$ orbits -- of the hypermetric cone $HYP_8$ are computed. The first generalization considered is the hypermetric polytope $HYPP_n$ for which we give general algorithms and a description for $n\le 8$. Then we shortly consider generalizations to simplices of volume higher than $1$, hypermetric on graphs and infinite dimensional hypermetrics.