The representation cone of an interval order

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.

show abstract

The Representation Polyhedron of a Semiorder

Balof

Doignon

Fiorini

2011

Order

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

The representation cone of an interval order

Cited by 1 publication

References 2 publications

The Representation Polyhedron of a Semiorder

The Representation Polyhedron of a Semiorder

Contact Info

Product

Resources

About