Abstract
Zielonka's classic recursive algorithm for solving parity games is perhaps the simplest among the many existing parity game algorithms. However, its complexity is exponential, while currently the state-of-the-art algorithms have quasipolynomial complexity. Here, we present a modification of Zielonka's classic algorithm that brings its complexity down to $n^{O\left(\log\left(1+\frac{d}{\log n}\right)\right)}$, for parity games of size $n$ with $d$ priorities, in line with previous quasipolynomial-time solutions.
Highlights
A parity game is an infinite two-player game in which Even and her opponent Odd build an infinite path along the edges of a graph labelled with integer priorities
The main advantage of our recursive approach over previous quasipolynomialtime algorithms solving parity games lies in its simplicity: we perform a small modification of the straightforward Zielonka’s algorithm
Czerwinski et al [CDF+19] argue that previous quasipolynomial-time algorithms solving parity games [CJK+17, JL17, Leh18, FJdK+19] are instances of a so-called separator approach. They prove that every algorithm accomplishing this approach has to follow a structure of a universal tree, and they show a quasipolynomial lower bound for the size of such a tree— for the running time of the algorithm
Summary
A parity game is an infinite two-player game in which Even and her opponent Odd build an infinite path along the edges of a graph labelled with integer priorities. Even’s goal is for the highest priority seen infinitely often on this path to be even, while Odd tries to stop her. Parity games are a central tool in automata theory, logic, and their applications to verification. The model-checking problem for the modal μ calculus [EJS01] and the synthesis problem for LTL [PR89] reduce to solving parity games. Solutions to parity games have influenced work on ω-word automata translations [BL18, DJL19], linear optimisation [Fri11b, FHZ11] and stochastic planning [Fea10]. Key words and phrases: Parity Games, Quasipolynomial algorithm, Zielonka’s algorithm. ∗ Journal version of Parys [Par19] and Lehtinen, Schewe, and Wojtczak [LSW19]
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