Primary Facets Of Order Polytopes

TL;DR

This paper investigates the geometric structure of order polytopes related to various order relations, focusing on primary facet-defining inequalities with coefficients -1, 0, or 1, and classifies these for three types of order polytopes.

Contribution

It provides a classification of primary facet-defining inequalities for three order polytopes and explores the complexity of these inequalities for linear and weak order polytopes.

Findings

Classified all primary facet-defining inequalities for three order polytopes.

Identified the complexity of primary inequalities in linear and weak order polytopes.

Highlighted the difficulty of obtaining complete facet descriptions for these polytopes.

Abstract

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.