par Balof, Barry 
;Doignon, Jean-Paul ;Fiorini, Samuel
Référence Order, 30, 1, page (103-135)
Publication Publié, 2013
;Doignon, Jean-Paul ;Fiorini, Samuel Référence Order, 30, 1, page (103-135)
Publication Publié, 2013
Article révisé par les pairs
| Résumé : | Let a finite semiorder, or unit interval order, be given. When suitably defined, its numerical representations are the solutions of a system of linear inequalities. They thus form a convex polyhedron. We show that the facets of the representation polyhedron correspond to the noses and hollows of the semiorder. Our main result is to prove that the system defining the polyhedron is totally dual integral. As a consequence, the coordinates of the vertices and the components of the extreme rays of the polyhedron are all integral multiples of a common value. Total dual integrality is in turn derived from a particular property of the oriented cycles in the directed graph of noses and hollows of a strictly upper diagonal step tableau. Our approach delivers also a new proof for the existence of the minimal representation of a semiorder, a concept originally discovered by Pirlot (Theory Decis 28:109-141, 1990). Finding combinatorial interpretations of the vertices and extreme rays of the representation polyhedron is left for future work. © 2011 Springer Science+Business Media B.V. |
