A Discrete Stratety Improvement Algorithm for Solving Parity Games

Marcin Jurdzinski
Jens Vöge

December 2000


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 $\mu$-calculus. Known strategy improvement algorithms, as proposed for stochastic games by Hoffman and Karp in 1966, and for discounted payoff games and parity games by Puri in 1995, work with real numbers and require solving linear programming instances involving high precision arithmetic. In the present algorithm for parity games these difficulties are avoided by the use of discrete vertex valuations in which information about the relevance of vertices and certain distances is coded. An efficient implementation is given for a strategy improvement step. Another advantage of the present approach is that it provides a better conceptual understanding and easier analysis of strategy improvement algorithms for parity games. However, so far it is not known whether the present algorithm works in polynomial time. The long standing problem whether parity games can be solved in polynomial time remains open

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