Solving Random Parity Games in Polynomial Time
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We consider the problem of solving random parity games. We prove 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. We further propose the SWCP (Self-Winning Cycles Propagation) algorithm and show that, when the degree is large enough, SWCP solves the game with high probability. Furthermore, the complexity of SWCP is polynomial O(| V|^2 + | V|| E|). The design of SWCP is based on the threshold for the appearance of particular types of cycles in the players' respective subgraphs. We further show that non-sparse games can be solved in time O(| V|) with high probability, and emit a conjecture concerning the hardness of the d=2 case.
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