Beyond Value Iteration for Parity Games: Strategy Iteration with Universal Trees

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 $O(mn^2\log3 n \log d)$ respectively per iteration.