On the k-Systems of a Simple Polytope

TL;DR

This paper introduces k-systems in simple polytopes' graphs, providing polynomial certificates for facets and objective functions, and characterizes acyclic orientations that produce abstract objective functions.

Contribution

It defines k-systems for simple polytopes, linking them to certificates and characterizations of abstract objective functions and acyclic orientations.

Findings

k-systems lead to polynomial certificates for facets and AOFs

Acyclic orientation induces an AOF iff it has a unique sink on every 2-face

Provides a new combinatorial framework for polytope face analysis

Abstract

A k-system of the graph G(P) of a simple polytope P is a set of induced subgraphs of G(P) that shares certain properties with the set of subgraphs induced by the k-faces of P. This new concept leads to polynomial-size certificates in terms of G(P) for both the set of vertex sets of facets as well as for abstract objective functions (AOF) in the sense of Kalai. Moreover, it is proved that an acyclic orientation yields an AOF if and only if it induces a unique sink on every 2-face.