New Bounds for the Integer Carathéodory Rank

Abstract

Given a pointed rational n-dimensional cone C, we obtain new parametric and asymptotic upper bounds for the integer Carathéodory rank CR(C), defined as the smallest integer k such that any integer vector in C can be expressed as a non-negative integer combination of at most k elements from the Hilbert basis of C. Firstly, we significantly improve previously known bounds on the integer Carathéodory rank in an asymptotic setting, where we only consider "most" integer vectors in C. Secondly, we show that the equality CR(C) = n, referred to as the integer Carathéodory property, holds in arbitrary dimension for cones that possess polyhedral representations with bimodular matrices. Furthermore, we obtain a new upper bound on CR(C) for simplicial cones represented by k-modular matrices with k ≤ n. Similar results are established for cones represented by generating sets.