Parity Games of Bounded Tree-Depth

Konrad Staniszewski

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arXiv · 2211.02926

Parity Games of Bounded Tree-Depth

Konrad Staniszewski

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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.

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Konrad Staniszewski. 2022-11-05. Parity Games of Bounded Tree-Depth. https://arxiv.org/abs/2211.02926

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