Primary Facets Of Order Polytopes

Mixture models on order relations play a central role in recent investigations of transitivity in binary choice data. In such a model, the vectors of choice probabilities are the convex combinations of the characteristic vectors of all order relations of a chosen type. The five prominent types of order relations are linear orders, weak orders, semiorders, interval orders and partial orders. For each of them, the problem of finding a complete, workable characterization of the vectors of probabilities is crucial---but it is reputably inaccessible. Under a geometric reformulation, the problem asks for a linear description of a convex polytope whose vertices are known. As for any convex polytope, a shortest linear description comprises one linear inequality per facet. Getting all of the facet-defining inequalities of any of the five order polytopes seems presently out of reach. Here we search for the facet-defining inequalities which we call primary because their coefficients take only the values -1, 0 or 1. We provide a classification of all primary, facet-defining inequalities of three of the five order polytopes. Moreover, we elaborate on the intricacy of the primary facet-defining inequalities of the linear order and the weak order polytopes

Facets of the weak order polytope derived from the induced partition projection

The weak order polytopes are studied in Gurgel and Wakabayashi [Discrete Math. 175 (1997), pp. 163-172], Gurgel and Wakabayashi [The Complete Pre-Order Polytope: Facets and Separation Problem, manuscript, 1996], and Fiorini and Fishburn [Weak order polytopes, submitted]. We make use of their natural, affine projection onto the partition polytopes to determine several new families of facets for them. It turns out that not all facets of partition polytopes are lifted into facets of weak order polytopes. We settle the cases of all facet-defining inequalities established for partition polytopes by Grötschel and Wakabayashi [Math. Programming, 47 (1990), pp. 367-387]. Our method, although rather simple, allows us to establish general families of facets which contain two particular cases previously requiring long proofs.SCOPUS: ar.jinfo:eu-repo/semantics/publishe