On Integer Images of Max-plus Linear Mappings

TL;DR

This paper investigates the problem of determining when max-plus linear mappings produce integer vectors, focusing on special cases that can be solved efficiently, with implications for job-scheduling problems.

Contribution

It introduces two polynomially solvable special cases for the problem of integer images in max-plus linear mappings, advancing towards a general polynomial solution.

Findings

Identified two special cases solvable in polynomial time

Connected the problem to job-scheduling applications

Provided insights into column adjustments for integer row maxima

Abstract

Let us extend the pair of operations (max,+) over real numbers to matrices in the same way as in conventional linear algebra. We study integer images of max-plus linear mappings. The question whether Ax (in the max-plus algebra) is an integer vector for at least one x has been studied for some time but polynomial solution methods seem to exist only in special cases. In the terminology of combinatorial matrix theory this question reads: is it possible to add constants to the columns of a given matrix so that all row maxima are integer? This problem has been motivated by attempts to solve a class of job-scheduling problems. We present two polynomially solvable special cases aiming to move closer to a polynomial solution method in the general case.