Parity games have witnessed several new quasi-polynomial algorithms since the breakthrough result of Calude et al. (STOC 2017). The combinatorial object underlying these approaches is a universal tree, as identified by Czerwi\'nski et al. (SODA 2019). By providing a quasi-polynomial lower bound on the size of universal trees, they have highlighted a barrier that must be overcome by all existing approaches to attain polynomial running time. This is due to the existence of worst case instances which force these algorithms to explore a large portion of the tree. As an attempt to overcome this barrier, we propose a strategy iteration framework which can be applied on any universal tree. It is at least as fast as its value iteration counterparts, while allowing one to take bigger leaps in the universal tree. Our main technical contribution is an efficient method for computing the least fixed point of 1-player games. This is achieved via a careful adaptation of shortest path algorithms to the setting of ordered trees. By plugging in the universal tree of Jurdzi\'nski and Lazi\'c (LICS 2017), or the Strahler universal tree of Daviaud et al. (ICALP 2020), we obtain instantiations of the general framework that take time $O(mn^2\log n\log d)$ and $O(mn^2\log^3 n \log d)$ respectively per iteration.

翻译：奇偶博弈自Calude等人（STOC 2017）的突破性成果以来，涌现出多种拟多项式时间算法。Czerwiński等人（SODA 2019）指出，这些方法背后的组合结构是通用树。通过证明通用树大小的拟多项式下界，他们揭示了所有现有方法在实现多项式运行时间时必须跨越的屏障——最坏情形实例迫使此类算法探索树中大量节点。为突破这一屏障，我们提出可应用于任意通用树的策略迭代框架。该框架在保持不低于价值迭代方法效率的同时，允许在通用树上实现更大步长的跳跃。本文主要技术贡献在于：针对单人博弈提出最小不动点的高效求解方法。通过将最短路径算法精巧适配至有序树场景，我们实现了这一目标。若采用Jurdziński与Lazić（LICS 2017）的通用树或Daviaud等人（ICALP 2020）的Strahler通用树，该通用框架的实例化版本每次迭代分别需要$O(mn^2\log n\log d)$和$O(mn^2\log^3 n \log d)$时间。