Computation of Atomic Fibers of Z-Linear Maps

TL;DR

This paper introduces a new algorithm for computing atomic fibers of Z-linear maps, which are fundamental in understanding the structure of integer solutions to linear systems, with practical applications demonstrated.

Contribution

The paper presents a novel algorithm specifically designed to compute atomic fibers of Z-linear maps, advancing computational methods in integer linear algebra.

Findings

Algorithm successfully computes atomic fibers in various examples

Computational efficiency improvements over previous methods

Applications demonstrated in related integer programming problems

Abstract

For given matrix $A\in\Z^{d\times n}$, the set $P_{b}=\{z:Az=b,z\in\Z^n_+\}$ describes the preimage or fiber of $b\in\Z^d$ under the $\Z$-linear map $f_A:\Z^n_+\to\Z^d$, $x\mapsto Ax$. The fiber $P_{b}$ is called atomic, if $P_{b}=P_{b_1}+P_{b_2}$ implies $b=b_1$ or $b=b_2$. In this paper we present a novel algorithm to compute such atomic fibers. An algorithmic solution to appearing subproblems, computational examples and applications are included as well.