Computing extreme rays of the metric cone for seven points
European Journal of Combinatorics
How good are convex hull algorithms?
Computational Geometry: Theory and Applications
On Skeletons, Diameters and Volumes of Metric Polyhedra
Selected papers from the 8th Franco-Japanese and 4th Franco-Chinese Conference on Combinatorics and Computer Science
Geometry of Cuts and Metrics
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We consider convex polyhedra with applications to well-known combinatorial optimization problems: the metric polytope mn and its relatives. For n 驴 6 the description of the metric polytope is easy as mn has at most 544 vertices partitioned into 3 orbits; m7 - the largest previously known instance - has 275 840 vertices but only 13 orbits. Using its large symmetry group, we enumerate orbitwise 1 550 825 600 vertices of the 28-dimensional metric polytope m8. The description consists of 533 orbits and is conjectured to be complete. The orbitwise incidence and adjacency relations are also given. The skeleton of m8 could be large enough to reveal some general features of the metric polytope on n nodes. While the extreme connectivity of the cuts appears to be one of the main features of the skeleton of mn, we conjecture that the cut vertices do not form a cut-set. The combinatorial and computational applications of this conjecture are studied. In particular, a heuristic skipping the highest degeneracy is presented.