Quasi-semi-metrics, Oriented Multi-cuts and Related Polyhedra

The notions of directed distances, quasi-metrics and oriented multi-cuts are generalizations of the notions of distances, metrics and cuts, which are central objects in graph theory and combinatorial optimization. Define a quasi-semi-metric on X as a function d from X2 to R+, such that for all x, y, z ∈ X , d(x, y) ≥ 0, d(x, x) = 0, and d(x, y) ≤ d(x, z) + d(z, y). If the first inequality is strict for all x 6= y, then d is called a quasi-metric. If for all x, y ∈ X , d(x, y) = d(y, x), then d is called a semi-metric and a metric, respectively. Given a partition (S1, . . . , Sq) (q ≥ 2) of X = {1, . . . , n}, a quasi-semi-metric δ(S1, . . . , Sq) is called an oriented multi-cut if δ(S1, . . . , Sq)(i, j) = 1 for i ∈ Sα , j ∈ Sβ , α < β, and δ(S1, . . . , Sq)(i, j) = 0, otherwise. After short review of general quasi-semi-metrics we consider, for small values of n, the cone and the polytope of all quasi-semi-metrics on X = {1, . . . , n} and the cone and the polytope generated by all oriented multi-cuts on X . We list the facets and generators for these polyhedra and tables of their adjacencies and incidences. We study the 1-skeleton graphs and the ridge graphs of these polyhedra: the number of nodes and edges of these graphs, their diameters, adjacency conditions, inclusions among these graphs and their restrictions on some orbits of nodes. Finally, we compare the results obtained for oriented case with similar results for the symmetric case (see [6–8, 10]). All computations were done using the programs cdd of [13]. The following notations will be used: