Polyhedra with the Integer Caratheodory Property

TL;DR

This paper proves that certain classes of polyhedra, including matroid base polytopes and those defined by TU matrices, satisfy the Integer Carathéodory Property, which relates to expressing integer vectors as convex combinations of affinely independent integer vectors.

Contribution

It establishes that matroid base polytopes and TU matrix-defined polyhedra satisfy the Integer Carathéodory Property, expanding understanding of polyhedral integer decompositions.

Findings

Matroid base polytopes satisfy the Integer Carathéodory Property.

Polyhedra defined by TU matrices satisfy the property.

Projections of these polyhedra also satisfy the property.

Abstract

A polyhedron P has the Integer Caratheodory Property if the following holds. For any positive integer k and any integer vector w in kP, there exist affinely independent integer vectors x_1,...,x_t in P and positive integers n_1,...,n_t such that n_1+...+n_t=k and w=n_1x_1+...+n_tx_t. In this paper we prove that if P is a (poly)matroid base polytope or if P is defined by a TU matrix, then P and projections of P satisfy the integer Caratheodory property.