A subexporiential lower bound for the Random Facet algorithm for Parity Games

Parity Games form an intriguing family of infinite duration games wliose solution is equivalent to the solution of important problems in automatic verification and automata theory. They also form a very natural subclass of Deterministic Mean Payoff Games, which in turn is a very natural subclass of turn-based Stochastic Mean Payoff Games. It is a major open problem whether these game families cam be solved in polynomial time. The currently theoretically fastest algorithms for the solution of all these games are adaptations of the randomized algorithms of Kalai and of Matoušek, Sharir and Welzl for LP-type problems, an abstract generalization of linear pro-gramming. The expected rutming time of both algorithms is subexponential in the size of the game, i.e., 2^O(√nlogn) where n is the number of vertices in the game. We focus in this paper on the algorithm of Matoušek, Sharir and Welzl and refer to it as the Random Facet algorithm. Matoušek constructed a family of abstract optimization problems such that the expected running time of the Random Facet algorithm, when run on a random instance from this family, is close to the subexponential upper bound given above. This shows that in the abstract setting, 2^O(√ nlogn) the upper bound on the complexity of the Random Facet algorithm is essentially tight. It is not known, however, whether the abstract optimization problems constructed by Matoušek correspond to games of any of the families mentioned above. There was some hope, therefore, that the Random Facet algorithm, when applied to, say, parity games, may run in polynomial time. We show, that this, unfortunately, is not the case by constructing explicit parity games on which the expected running time of the Random Facet algorithm is close to the subexponential upper bound. The games we use mimic the behavior of a randomized counter. They are also the first explicit LP-type problems on which the Random Facet algorithm is not polynomial.