# Deciding parity games in quasipolynomial time

@article{Calude2017DecidingPG, title={Deciding parity games in quasipolynomial time}, author={Cristian S. Calude and Sanjay Jain and Bakhadyr Khoussainov and Wei Li and Frank Stephan}, journal={Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing}, year={2017} }

It is shown that the parity game can be solved in quasipolynomial time. The parameterised parity game - with n nodes and m distinct values (aka colours or priorities) - is proven to be in the class of fixed parameter tractable (FPT) problems when parameterised over m. Both results improve known bounds, from runtime nO(√n) to O(nlog(m)+6) and from an XP-algorithm with runtime O(nΘ(m)) for fixed parameter m to an FPT-algorithm with runtime O(n5)+g(m), for some function g depending on m only. As…

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## 175 Citations

An ordered approach to solving parity games in quasi-polynomial time and quasi-linear space

- Computer Science, MathematicsInternational Journal on Software Tools for Technology Transfer
- 2019

This work provides a first implementation for a quasi-polynomial algorithm, test it on small examples, and provides a number of side results, including minor algorithmic improvements, and a complexity index associated to the approach, which is compared to the recently proposed register index.

An ordered approach to solving parity games in quasi polynomial time and quasi linear space

- Computer ScienceSPIN
- 2017

A first implementation for a quasi-polynomial algorithm is provided, and a number of side results are provided, including minor algorithmic improvements, a quasi bi-linear complexity in the number of states and edges for a fixed number of colours, and matching lower bounds for the algorithm of Calude et al.

BDD-based parity game solving

- Computer Science
- 2018

Zielonka’s BDD-based algorithm beats the BDDbased Priority Promotion algorithm by a small margin for games that are characteristic of practical verification problems, and for games with a large number of distinct priorities, Priority Promotion scales better.

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

- Computer ScienceMFCS
- 2019

A small modification of the Zielonka's algorithm is proposed, which ensures that the running time is at most quasi-polynomial, and it is hoped that this algorithm, after further optimizations, can lead to an algorithm that shares the good performance of the ziggurat's algorithm on typical inputs, while reducing the worst-case complexity on difficult inputs.

Solving Random Parity Games in Polynomial Time

- Computer ScienceArXiv
- 2020

It is proved that parity games exibit a phase transition threshold above $d_P$, so that when the degree of the graph that defines the game has a degree $d > d_P$ then there exists a polynomial time algorithm that solves the game with high probability when the number of nodes goes to infinity.

An Optimal Value Iteration Algorithm for Parity Games

- Computer ScienceArXiv
- 2018

The technical result of this paper is to show that the latter construction is asymptotically tight: universal trees have at least quasipolynomial size, suggesting that the succinct progress measure algorithm of Jurdzi\'nski and Lazi\'c is in this framework optimal, and that the polynomial time algorithm for parity games is hiding someplace else.

Improving parity games in practice

- Computer ScienceAnn. Math. Artif. Intell.
- 2021

This work deeply revisit the implementation of Zielonka’s recursive algorithm (RE) by dealing with the use of specific data structures and programming languages such as Scala, Java, C++, and Go, and shows that these choices are successful.

A Quasi-Polynomial Black-Box Algorithm for Fixed Point Evaluation

- Computer Science, MathematicsCSL
- 2021

Following a recent development for parity games, it is shown here that a quasi-polynomial number of queries is sufficient, namely nlg(d/ lg n)+O(1), an abstract version of several algorithms proposed recently by a number of authors, which involve (implicitly or explicitly) the structure of a universal tree.

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A first implementation for a quasi-polynomial algorithm is provided, and a number of side results are provided, including minor algorithmic improvements, a quasi bi-linear complexity in the number of states and edges for a fixed number of colours, and matching lower bounds for the algorithm of Calude et al.

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