The Representation Polyhedron of a Semiorder

Abstract

Abstract. 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.

Extending Partial Representations of Proper and Unit Interval Graphs

“…Our algorithm makes use of several structural properties of the set of all representations. We note that structural properties of the polyhedron of our linear program, in the case where all lower bounds equal zero and there are no upper bounds, have been considered in several papers in the context of semiorders [29,30].…”

Unit representation of semiorders II: The general case

“…The latter is expressed in terms of noses, a notion introduced in Pirlot (1990Pirlot ( , 1991, together with that of hollows. These have proved useful in the study of the unit representations of finite semiorders (Balof et al, 2013, Doignon, 1988, Pirlot, 1990, 1991. The results in this section have appeared in Section 3.2 of Bouyssou and Pirlot (2020b).…”

“…The latter also clarifies the relationships with Beja and Gilboa's characterization. Note that the conditions that have to be added to Debreu-separability for obtaining either strict or nonstrict representations use the notions of noses and hollows that were fruitful in the study of finite semiorders (Balof et al, 2013, Doignon, 1988, Pirlot, 1990, 1991. The paper is organized as follows.…”