# Parity games and universal graphs

@article{Colcombet2018ParityGA, title={Parity games and universal graphs}, author={Thomas Colcombet and Nathana{\"e}l Fijalkow}, journal={ArXiv}, year={2018}, volume={abs/1810.05106} }

This paper is a contribution to the study of parity games and the recent constructions of three quasipolynomial time algorithms for solving them. We revisit a result of Czerwi\'nski, Daviaud, Fijalkow, Jurdzi\'nski, Lazi\'c, and Parys witnessing a quasipolynomial barrier for all three quasipolynomial time algorithms. The argument is that all three algorithms can be understood as constructing a so-called separating automaton, and to give a quasipolynomial lower bond on the size of separating…

## 9 Citations

Universal trees grow inside separating automata: Quasi-polynomial lower bounds for parity games

- Mathematics, Computer ScienceSODA
- 2019

The technical highlights are a quasi-polynomial lower bound on the size of universal ordered trees and a proof that every separating safety automaton has a universal tree hidden in its state space.

The complexity of mean payoff games using universal graphs

- Mathematics, Computer ScienceArXiv
- 2018

It is shown that separating automata do not yield a quasipolynomial algorithm for solving mean payoff games, and tight bounds on the complexity of algorithms in this class are proved.

Value Iteration Using Universal Graphs and the Complexity of Mean Payoff Games

- Mathematics, Computer ScienceMFCS
- 2020

It is shown that the linear dependence in the exponent in the number k of weights implies that universal graphs do not yield a quasipolynomial time algorithm for solving mean payoff games, implying that tight bounds on the complexity of algorithms formean payoff games using universal graphs are proved.

Universal Graphs and Good for Games Automata: New Tools for Infinite Duration Games

- Computer ScienceFoSSaCS
- 2019

A self contained presentation of a recent breakthrough in the theory of infinite duration games: the existence of a quasipolynomial time algorithm for solving parity games and two new notions: good for small games automata and universal graphs.

A symmetric attractor-decomposition lifting algorithm for parity games

- Mathematics, Computer ScienceArXiv
- 2020

The behaviour of the generic attractor-based algorithm of Jurdzinski and Morvan (2020) can be reproduced by a specific deceleration of the authors' symmetric lifting algorithm, in which some of the information collected by the algorithm is repeatedly discarded, further strengthening the ties between all known quasi-polynomial algorithms to date.

New Algorithms for Combinations of Objectives using Separating Automata

- Computer ScienceGandALF
- 2021

It is shown that separating automata is a powerful tool for constructing algorithms solving games with combinations of objectives, and two new algorithms are constructed for disjunctions of parity and mean payoff objectives, matching the best known complexity.

Alternating Weak Automata from Universal Trees

- Computer Science, MathematicsCONCUR
- 2019

Any slightly better translation from alternating parity automata on infinite words to alternating weak automata would lead to algorithms for solving parity games that are asymptotically faster in the worst case than the current state of the art, and hence would yield a significant breakthrough.

The Theory of Universal Graphs for Infinite Duration Games

- Computer ScienceArXiv
- 2021

The theory of universal graphs is developed, with two goals: showing an equivalence and normalisation result between different recently introduced related models, and constructing generic value iteration algorithms for any positionally determined objective.

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