The tropical shadow-vertex algorithm solves mean payoff games in polynomial time on average

TL;DR

This paper presents a novel tropical shadow-vertex algorithm that efficiently solves mean payoff games in polynomial time on average, leveraging tropical geometry and combinatorial optimization techniques.

Contribution

It introduces a tropical analogue of the shadow-vertex algorithm, enabling polynomial-time solutions for mean payoff games under certain distributional assumptions.

Findings

Algorithm solves mean payoff games in polynomial time on average

Uses tropical geometry and Plücker relations for computation

Reduces pivoting rule to optimal assignment problems

Abstract

We introduce an algorithm which solves mean payoff games in polynomial time on average, assuming the distribution of the games satisfies a flip invariance property on the set of actions associated with every state. The algorithm is a tropical analogue of the shadow-vertex simplex algorithm, which solves mean payoff games via linear feasibility problems over the tropical semiring $(\mathbb{R} \cup \{-\infty\}, \max, +)$. The key ingredient in our approach is that the shadow-vertex pivoting rule can be transferred to tropical polyhedra, and that its computation reduces to optimal assignment problems through Pl\"ucker relations.