The Biorder Polytope

Abstract

Biorders, also called Ferrers relations, formalize Guttman scales. Irreflexive biorders on a set are exactly the interval orders on that set. The biorder polytope is the convex hull of the characteristic matrices of biorders. Its definition is thus similar to the definition of other order polytopes, the linear ordering polytope being the proeminent example. We investigate the combinatorial structure of the biorder polytope, thus obtaining a complete linear description in a specific case, and the automorphism group in all cases. Moreover, a class of facet-defining inequalities defined from weighted graphs is thoroughly analyzed. A weighted generalization of stability-critical graphs is presented, which leads to new facets even for the well-studied linear ordering polytope.