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

Combinatorics of finite ordered sets: order polytopes and poset entropy

The thesis focuses on two open problems on finite partially ordered sets: the structure of order polytopes and the approximation of the number of linear extensions of a poset by mean of graph entropy. The polytopes considered here are the linear ordering polytope, the semiorder polytope, the interval order polytope, the partial order polytope and also a generalisation of the linear ordering polytope: the linear extension polytope of a fixed poset P. Various results on the structure of theses polytopes are proved in the first part of the thesis. In the second part of the thesis, we improve the existing bounds linking the entropy of the incomparability graph of the poset P and its number of linear extension.Le but de la thèse est d'étudier deux problèmes ouverts sur les ensembles ordonnés finis: la structure des polytopes d'ordre et l'approximation du nombre d'extensions linéaires d'un ordre partiel au moyen de la notion d'entropie de graphe. Les polytopes considérés sont le polytope des ordres totaux, le polytope des semiordres, le polytope des ordres d'intervalles, le polytope des ordres partiels, ainsi qu'une généralisation du polytope des ordres totaux: le polytope des extensions linéaires d'un ensemble ordonné fixé P. Des résultats sur la structure de ces polytopes sont présentés dans la première partie de la thèse. Dans la deuxième partie de la thèse, nous améliorons les bornes existantes liant l'entropie du graphe d'incomparabilité d'un ordre partiel et son nombre d'extensions linéaires.Doctorat en Sciencesinfo:eu-repo/semantics/nonPublishe

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.SCOPUS: ar.jSCOPUS: ar.jinfo:eu-repo/semantics/publishe

The Linear Extension Polytope of a Poset

Let P be a finite poset. By definition, the linear extension polytope of P has as vertices the characteristic vectors of all linear extensions of P. In case P is an antichain, it is the linear ordering polytope. The linear extension polytope appears in combinatorial optimization in the context of scheduling with precedence constraints, see e.g. [A. Schulz, Polytopes and Scheduling, Phd Thesis, TU Berlin, 1996]. It seems also relevant to order theory, being similar in spirit to other constructions such as the linear extension graph, see e.g. [M. Massow, Linear extension graphs and linear extension diameter, PhD thesis, TU Berlin, 2009]. In this work, we relate the combinatorial properties of the poset P to the polyhedral structure of its linear extension polytope. Of particular interest is a natural relaxation of the linear extension polytope. We prove that the relaxation is exact in case P is a width-2 poset, and formulate a conjecture stating exactly when the relaxation is exact.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Poset Entropy Versus Number of Linear Extensions: The Width-2 Case

Kahn and Kim (J. Comput. Sci. 51, 3, 390–399, 1995) have shown that for a finite poset P, the entropy of the incomparability graph of P (normalized by multiplying by the order of P) and the base-2 logarithm of the number of linear extensions of P are within constant factors from each other. The tight constant for the upper bound was recently shown to be 2 by Cardinal et al. (Combinatorica 33, 655–697, 2013). Here, we refine this last result in case P has width 2: we show that the constant can be replaced by 2−ε if one also takes into account the number of connected components of size 2 in the incomparability graph of P. Our result leads to a better upper bound for the number of comparisons in algorithms for the problem of sorting under partial information.SCOPUS: ar.jinfo:eu-repo/semantics/publishe