Parity Games: Zielonka's Algorithm in Quasi-Polynomial Time

TL;DR

This paper introduces a modified Zielonka's recursive algorithm for parity games that achieves quasi-polynomial time complexity, combining simplicity and practical efficiency with improved theoretical guarantees.

Contribution

A small modification to Zielonka's algorithm is proposed, ensuring quasi-polynomial worst-case running time while maintaining simplicity and practical performance.

Findings

The modified algorithm runs in quasi-polynomial time.

It retains the simplicity and efficiency of Zielonka's original algorithm.

Potential for further optimization to improve worst-case complexity.

Abstract

Calude, Jain, Khoussainov, Li, and Stephan (2017) proposed a quasi-polynomial-time algorithm solving parity games. After this breakthrough result, a few other quasi-polynomial-time algorithms were introduced; none of them is easy to understand. Moreover, it turns out that in practice they operate very slowly. On the other side there is the Zielonka's recursive algorithm, which is very simple, exponential in the worst case, and the fastest in practice. We combine these two approaches: we propose a small modification of the Zielonka's algorithm, which ensures that the running time is at most quasi-polynomial. In effect, we obtain a simple algorithm that solves parity games in quasi-polynomial time. We also hope that our algorithm, after further optimizations, can lead to an algorithm that shares the good performance of the Zielonka's algorithm on typical inputs, while reducing the worst-case complexity on difficult inputs.