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

2021-08-30

International Symposium on Mathematical Foundations of Computer Science

AI Summary

Existing generic-tree algorithms for parity games suffer from high tree-traversal overhead on worst-case instances, preventing polynomial-time implementation. Method: We propose the first strategy-iteration framework built upon arbitrary generic trees—introducing strategy iteration, rather than value iteration, to generic-tree structures. It computes large-step updates by efficiently solving minimal fixed points of single-player subgames, thereby avoiding deep tree traversals. Integrated with a generic-tree-adapted shortest-path algorithm and instantiated via Jurdziński–Lazić or Strahler trees, our framework achieves per-iteration time complexity of either $O(mn^2 log n log d)$ or $O(mn^2 log^3 n log d)$. Contribution/Results: The method significantly reduces practical convergence steps and breaks the asymptotic limitations of prior quasi-polynomial algorithms, establishing the first generic-tree-based strategy-iteration approach with improved theoretical and empirical efficiency.

Abstract

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 proving a quasi-polynomial lower bound on the size of a universal tree, 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 $O(mn^2log nlog d)$ and $O(mn^2log^3 n log d)$ respectively per iteration.

Problem

• Overcome quasi-polynomial barrier in parity games
• Propose strategy iteration for universal trees
• Compute least fixed point efficiently in 1-player games

Innovation

• Strategy iteration framework on universal trees
• Efficient least fixed point computation method
• Adaptation of shortest path algorithms