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. (2017). The central combinatorial object underlying these approaches is a universal tree, as identified by Czerwi\'nski et al. (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 runtime. 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. Value iteration - asymptotically the fastest known algorithm for parity games - is a repeated application of operators associated with arcs in the game graph to obtain the least fixed point. Our main technical contribution is an efficient method for computing the least fixed point of operators associated with arcs in a strategy subgraph. 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 (2017), or the Strahler universal tree of Daviaud et al. (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.