On the Face Lattice of the Metric Polytope

In this paper we study enumeration problems for polytopes arising from combinatorial optimization problems. While these polytopes turn out to be quickly intractable for enumeration algorithms designed for general polytopes, tailor-made algorithms using their rich combinatorial features can exhibit strong performances. The main engine of these combinatorial algorithms is the use of the large symmetry group of combinatorial polytopes. Specifically we consider a polytope with applications to the well-known max-cut and multicommodity flow problems: the metric polytope mn on n nodes. We prove that for n ≥ 9 the faces of codimension 3 of the metric polytope are partitioned into 15 orbits of its symmetry group. For n ≤ 8, we describe additional upper layers of the face lattice of mn. In particular, using the list of orbits of high dimensional faces of m8, we prove that the description of m8 given in [9] is complete with 1 550 825 000 vertices and that the Laurent-Poljak conjecture [14] holds for n ≤ 8. Many vertices of m9 are computed and additional results on the structure of the metric polytope are presented. Computational issues for the orbitwise face and vertex enumeration algorithms are also discussed.