An algorithm to describe the solution set of any tropical linear system $A\odot x=B\odot x$

TL;DR

This paper presents an algorithm that explicitly describes all solutions to any tropical linear system by converting it into a small set of classical linear systems involving only bivariate equations and inequalities.

Contribution

It introduces the concept of compatibility to reduce the problem to a finite set of simple systems and develops methods to solve these using Gaussian elimination and sub-specialization.

Findings

The algorithm explicitly describes the entire solution set.

The conversion results in a small, manageable number of classical systems.

Solutions are obtained through adapted Gaussian elimination techniques.

Abstract

An algorithm to give an explicit description of all the solutions to any tropical linear system $A\odot x=B\odot x$ is presented. The given system is converted into a finite (rather small) number $p$ of pairs $(S,T)$ of classical linear systems: a system $S$ of equations and a system $T$ of inequalities. The notion, introduced here, that makes $p$ small, is called compatibility. The particular feature of both $S$ and $T$ is that each item (equation or inequality) is bivariate, i.e., it involves exactly two variables; one variable with coefficient $1$, and the other one with $-1$. $S$ is solved by Gaussian elimination. We explain how to solve $T$ by a method similar to Gaussian elimination. To achieve this, we introduce the notion of sub--special matrix. The procedure applied to $T$ is, therefore, called sub--specialization.