New Bounds for the Integer Carath\'{e}odory Rank

TL;DR

This paper improves bounds on the integer Carathéodory rank for rational cones, providing new asymptotic and exact bounds based on the cone's integral structure, especially for small minors.

Contribution

It significantly refines the upper bounds for the asymptotic Carathéodory rank and establishes exact values and bounds for cones with small minors, advancing understanding in polyhedral combinatorics.

Findings

Improved upper bounds for asymptotic Carathéodory rank.

Exact value of Carathéodory rank for cones with minor 1 or 2.

Enhanced bounds for simplicial cones with small minors.

Abstract

Given a rational pointed $n$-dimensional cone $C$, we study the integer Carath\'{e}odory rank $\operatorname{CR}(C)$ and its asymptotic form $\operatorname{CR^{\rm a}}(C)$, where we consider ``most'' integer vectors in the cone. The main result significantly improves the previously known upper bound for $\operatorname{CR^{\rm a}}(C)$. We also study bounds on $\operatorname{CR}(C)$ in terms of $\Delta$, the maximal absolute $n\times n$ minor of the matrix given in an integral polyhedral representation of $C$. If $\Delta\in\lbrace 1,2\rbrace$, we show $\operatorname{CR}(C) = n$, and prove upper bounds for simplicial cones, improving the best known upper bound on $\operatorname{CR}(C)$ for $\Delta\leq n$.