# The complexity of mean payoff games using universal graphs

## TL;DR

This paper explores the complexity of solving mean payoff games, extending techniques from parity games, and introduces new algorithms with tight bounds, demonstrating limitations of current approaches.

## Contribution

It extends the concept of separating automata to mean payoff games and provides two new algorithms with proven bounds, showing their limitations.

## Key findings

- First algorithm depends on the largest weight N and runs in sublinear time in N.
- Second algorithm runs in polynomial time for fixed number of weights k.
- Lower bounds show the limitations of separating automata approach.

## Abstract

We study the computational complexity of solving mean payoff games. This class of games can be seen as an extension of parity games, and they have similar complexity status: in both cases solving them is in $\textbf{NP} \cap \textbf{coNP}$ and not known to be in $\textbf{P}$. In a breakthrough result Calude, Jain, Khoussainov, Li, and Stephan constructed in 2017 a quasipolynomial time algorithm for solving parity games, which was quickly followed by two other algorithms with the same complexity. Our objective is to investigate how these techniques can be extended to the study of mean payoff games. The starting point is the notion of separating automata, which has been used to present all three quasipolynomial time algorithms for parity games and gives the best complexity to date. The notion naturally extends to mean payoff games and yields a class of algorithms for solving mean payoff games. The contribution of this paper is to prove tight bounds on the complexity of algorithms in this class. We construct two new algorithms for solving mean payoff games. Our first algorithm depends on the largest weight $N$ (in absolute value) appearing in the graph and runs in sublinear time in $N$, improving over the previously known linear dependence in $N$. Our second algorithm runs in polynomial time for a fixed number $k$ of weights. We complement our upper bounds by providing in both cases almost matching lower bounds, showing the limitations of the separating automata approach. We show that we cannot hope to improve on the dependence in $N$ nor break the linear dependence in the exponent in the number $k$ of weights. In particular, this shows that separating automata do not yield a quasipolynomial algorithm for solving mean payoff games.