# Small Progress Measures for Solving Parity Games

@inproceedings{Jurdzinski2000SmallPM, title={Small Progress Measures for Solving Parity Games}, author={Marcin Jurdzinski}, booktitle={STACS}, year={2000} }

In this paper we develop a new algorithm for deciding the winner in parity games, and hence also for the modal µ-calculus model checking. The design and analysis of the algorithm is based on a notion of game progress measures: they are witnesses for winning strategies in parity games. We characterize game progress measures as pre-fixed points of certain monotone operators on a complete lattice. As a result we get the existence of the least game progress measures and a straightforward way to…

## 443 Citations

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

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- 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.

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- Computer ScienceCAV
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A discrete strategy improvement algorithm is given for constructing winning strategies in parity games, thereby providing also a new solution of the model-checking problem for the modal μ-calculus.…

A Super-Polynomial Lower Bound for the Parity Game Strategy Improvement Algorithm as We Know it

- Computer ScienceArXiv
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A new lower bound for the discrete strategy improvement algorithm for solving parity games due to Voege and Jurdziski is presented, answering in the negative the long-standing question whether this algorithm runs in polynomial time.

The Fixpoint-Iteration Algorithm for Parity Games

- Computer ScienceGandALF
- 2014

This paper studies the effect of employing the most straight-forward mu-calculus model checking algorithm: fixpoint iteration, and shows that particular exponential-space strategies which are eventually-positional can be extracted from them, and are shown to be positional winning strategies.

An Exponential Lower Bound for the Parity Game Strategy Improvement Algorithm as We Know it

- Computer Science2009 24th Annual IEEE Symposium on Logic In Computer Science
- 2009

A family of games on which the discrete strategy improvement algorithm for solving parity games due to Voege and Jurdzinski requires exponentially many strategy iterations is outlined, answering in the negative the long-standing question whether this algorithm runs in polynomial time.

Local Strategy Improvement for Parity Game Solving

- Computer ScienceGANDALF
- 2010

A local strategy improvement algorithm which explores the game graph on-the-fly whilst performing the improvement steps and can outperform existing global strategy improvement algorithms by several orders of magnitude.

Two Local Strategy Iteration Schemes for Parity Game Solving

- Computer ScienceInt. J. Found. Comput. Sci.
- 2012

It turns out that local strategy iteration can outperform these others significantly and be compared empirically with existing global strategy iteration algorithms and the currently only other local algorithm for solving parity games.

TWO LOCAL STRATEGY IMPROVEMENT SCHEMES FOR PARITY GAME SOLVING

- Computer Science
- 2010

Two local strategy improvement algorithms which explore the game graph on-the-fly whilst performing the improvement steps and can outperform existing global strategy improvement algorithm for solving parity games by several orders of magnitude.

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