The Projected Faces Property and Polyhedral Relations

TL;DR

This paper explores the relationship between the projected faces property and affinely generated polyhedral relations, providing a characterization and establishing their equivalence in the context of extended formulations of polytopes.

Contribution

It proves the equivalence of the projected faces property and affinely generated polyhedral relations, offering a new characterization relevant to extended formulations.

Findings

The projected faces property is necessary for certain extended formulations.

Affinely generated polyhedral relations are equivalent to the projected faces property.

Provides a characterization of the projected faces property in polyhedral theory.

Abstract

Margot (1994) in his doctoral dissertation studied extended formulations of combinatorial polytopes that arise from "smaller" polytopes via some composition rule. He introduced the "projected faces property" of a polytope and showed that this property suffices to iteratively build extended formulations of composed polytopes. For the composed polytopes, we show that an extended formulation of the type studied in this paper is always possible only if the smaller polytopes have the projected faces property. Therefore, this produces a characterization of the projected faces property. Affinely generated polyhedral relations were introduced by Kaibel and Pashkovich (2011) to construct extended formulations for the convex hull of the images of a point under the action of some finite group of reflections. In this paper we prove that the projected faces property and affinely generated polyhedral relation are equivalent conditions.