Solving Random Parity Games in Polynomial Time

TL;DR

This paper demonstrates that random parity games can be solved in polynomial time above a certain degree threshold, introduces the SWCP algorithm, and analyzes complexity and phase transitions in game solvability.

Contribution

It proves a phase transition threshold for polynomial solvability of random parity games and introduces the SWCP algorithm with polynomial complexity.

Findings

Parity games exhibit a phase transition at degree $d_P$.

SWCP algorithm solves large-degree games with high probability.

Non-sparse games can be solved in linear time with high probability.

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 $O\Big(|{\cal V}|^2 + |{\cal V}||{\cal E}|\Big)$. 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.