Solving Random Parity Games in Polynomial Time

2020/07/16 by Richard Combes, Combes, Richard, Mikaël Touati +1
Computer Science · #Advanced Graph Theory Research #Artificial Intelligence in Games #Complexity and Algorithms in Graphs #FOS: Computer and information sciences #Logic in Computer Science (cs.LO) #Machine Learning (stat.ML)

paper · pdf · doi:10.48550/arxiv.2007.08387

openalex publication_date 2020/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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(|\cal V|² + |\cal V||\cal 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(|\cal V|) with high probability, and emit a conjecture concerning the hardness of the d=2 case.

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