Tropical linear-fractional programming and parametric mean payoff games

TL;DR

This paper develops a tropical analogue of linear-fractional programming, connecting it to parametric mean payoff games, and introduces a Newton-like algorithm for solving these problems.

Contribution

It introduces a novel tropical linear-fractional programming framework and a new algorithm based on mean payoff games for solving it.

Findings

Optimality and unboundedness can be certified via strategies with infinitesimal properties.

A Newton-like algorithm is designed by reducing to auxiliary mean payoff game problems.

The approach extends tropical polyhedra applications in static analysis.

Abstract

Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, tropical analogues of classical linear programming results need to be developed. This motivation leads us to study the tropical analogue of the classical linear-fractional programming problem. We construct an associated parametric mean payoff game problem, and show that the optimality of a given point, or the unboundedness of the problem, can be certified by exhibiting a strategy for one of the players having certain infinitesimal properties (involving the value of the game and its derivative) that we characterize combinatorially. We use this idea to design a Newton-like algorithm to solve tropical linear-fractional programming problems, by reduction to a sequence of auxiliary mean payoff game problems.