The complexity of mean payoff games using universal graphs.

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. It has recently been shown that the notion of universal graphs captures the combinatorial structure behind all three algorithms and gives a unified presentation together with the best complexity to date. In this paper we investigate how these techniques can be extended, and more specifically we give upper and lower bounds on the complexity of algorithms using universal graphs for solving mean payoff games. We construct two new algorithms each focussing on one of the two following parameters: the largest weight $N$ (in absolute value) appearing in the graph and the number $k$ of weights. Our first algorithm improves the best known complexity by reducing the dependence on $N$ from $N$ to $N^{1 - 1/n}$, where $n$ is the number of vertices in the graph. Our second algorithm runs in polynomial time for a fixed number $k$ of weights, more specifically in $O(m n^k)$, where $m$ is the number of edges in the graph. We complement our upper bounds by providing in both cases almost matching lower bounds, showing the limitations of the approach. We show that using universal graphs we cannot hope to improve on the $N^{1 - 1/n}$ dependence in $N$ nor break the $O(n^{\Omega(k)})$ barrier. In particular, universal graphs do not yield a quasipolynomial algorithm for solving mean payoff games.