2022/11/06 by Iskander Aliev, Martin Henk, Aliev, Iskander +7 · 1 citation
Computer Science · Engineering · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Optimization and Control (math.OC) #graph theory and CDMA systems
paper · doi:10.48550/arxiv.2211.03150
openalex publication_date 2022/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a rational pointed n-dimensional cone C, we study the integer Carathéodory rank CR(C) and its asymptotic form \operatornameCR\rm a(C), where we consider ``most'' integer vectors in the cone. The main result significantly improves the previously known upper bound for \operatornameCR\rm a(C). We also study bounds on CR(C) in terms of Δ, the maximal absolute n× n minor of the matrix given in an integral polyhedral representation of C. If Δ∈\lbrace 1,2\rbrace, we show CR(C) = n, and prove upper bounds for simplicial cones, improving the best known upper bound on CR(C) for Δ≤ n.