The Representation Polyhedron of a Semiorder

Let a finite semiorder, or unit interval order, be given. All its numerical representations (when suitably defined) 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 coordinates of the vertices and the components of the extreme rays of the polyhedron are all integral multiples of a common value. The result follows from the total dual integrality of the system defining the representation polyhedron. 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, a concept originally discovered by Pirlot (1990). Finding combinatorial interpretations of the vertices and extreme rays of the representation polyhedron is left for future work.