Substitution for minimizing/maximizing a tropical linear (fractional) programming

TL;DR

This paper introduces a strong polynomial substitution method for solving tropical linear (fractional) minimization and maximization problems, with applications in static analysis and game theory.

Contribution

A novel special substitution technique for tropical linear optimization that inherits strong polynomiality and handles fractional problems without special assumptions.

Findings

The substitution method is strongly polynomial and applicable to tropical linear minimization and maximization.

Tropicalizing Charnes-Cooper's transformation solves fractional linear problems in tropical algebra.

The method is illustrated with examples from the literature.

Abstract

Tropical polyhedra seem to play a central role in static analysis of softwares. These tropical geometrical objects play also a central role in parity games especially mean payoff games and energy games. And determining if an initial state of such game leads to win the game is known to be equivalent to solve a tropical linear optimization problem. This paper mainly focus on the tropical linear minimization problem using a special substitution method on the tropical cone obtained by homogenization of the initial tropical polyhedron. But due to a particular case which can occur in the minimization process based on substitution we have to switch on a maximization problem. Nevertheless, forward-backward substitution is known to be strongly polynomial. The special substitution developed in this paper inherits the strong polynomiality of the classical substitution for linear systems. This special substitution must not be confused with the exponential execution time of the tropical Fourier-Motzkin elimination. Tropical fractional minimization problem with linear objective functions is also solved by tropicalizing the Charnes-Cooper's transformation of a fractional linear program into a linear program developed in the usual linear algebra. Let us also remark that no particular assumption is made on the polyhedron of interest. Finally, the substitution method is illustrated on some examples borrowed from the litterature.