Parity Games of Bounded Tree-Depth

TL;DR

This paper demonstrates that solving parity games on graphs with bounded tree-depth can be efficiently done in logspace uniform AC^0, extending previous results on graph classes like bounded tree-width.

Contribution

It introduces shallow clique-width and shows that parity games on graphs with bounded tree-depth are solvable in logspace uniform AC^0, a significant complexity improvement.

Findings

Parity games on bounded tree-depth graphs are in logspace uniform AC^0.

Introduces shallow clique-width as a new graph parameter.

Provides a reduction from bounded tree-depth graphs to shallow clique-width graphs.

Abstract

The exact complexity of solving parity games is a major open problem. Several authors have searched for efficient algorithms over specific classes of graphs. In particular, Obdr\v{z}\'{a}lek showed that for graphs of bounded tree-width or clique-width, the problem is in $\mathrm{P}$, which was later improved by Ganardi, who showed that it is even in $\mathrm{LOGCFL}$ (with an additional assumption for clique-width case). Here we extend this line of research by showing that for graphs of bounded tree-depth the problem of solving parity games is in logspace uniform $\text{AC}^0$. We achieve this by first considering a parameter that we obtain from a modification of clique-width, which we call shallow clique-width. We subsequently provide a suitable reduction.