SIAM Journal on Computing, Ahead of Print.
It is shown that the parity game can be solved in quasi-polynomial time. The parameterized parity game---with $n$ nodes and $m$ distinct values (a.k.a. colors or priorities)---is proven to be in the class of fixed parameter tractable problems when parameterized over $m$. Both results improve known bounds, from runtime $n^{O(\sqrt{n})}$ to $O(n^{\log(m)+6})$ and from an XP algorithm with runtime $O(n^{\Theta(m)})$ for fixed parameter $m$ to a fixed parameter tractable algorithm with runtime $O(n^5+2^{m\log(m)+6m})$. As an application, it is proven that colored Muller games with $n$ nodes and $m$ colors can be decided in time $O((m^m \cdot n)^5)$; it is also shown that this bound cannot be improved to $2^{o(m \cdot \log(m))} \cdot n^{O(1)}$ in the case that the exponential time hypothesis is true. Further investigations deal with memoryless Muller games and multidimensional parity games.

中文翻译：

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拟多项式时间内的奇偶校验游戏

《 SIAM计算杂志》，预印本。
结果表明，奇偶博弈可以在拟多项式时间内求解。参数化的奇偶游戏-具有$ n $个节点和$ m $个不同的值（又称颜色或优先级）-被证明在参数化超过$ m $时属于固定参数易处理的问题类别。从运行时$ n ^ {O（\ sqrt {n}）} $到$ O（n ^ {\ log（m）+6}）$以及运行时$ O（n将固定参数$ m $的^ {\ Theta（m）}）$转换为运行时$ O（n ^ 5 + 2 ^ {m \ log（m）+ 6m}）$的固定参数可处理算法。作为一种应用，证明了具有$ n $个节点和$ m $个颜色的穆勒彩色游戏可以及时确定$ O（（m ^ m \ cdot n）^ 5）$; 还表明，在指数时间假设为真的情况下，不能将该边界提高到$ 2 ^ {o（m \ cdot \ log（m））} \ cdot n ^ {O（1）} $。