论文标题
在多项式时间求解随机的平价游戏
Solving Random Parity Games in Polynomial Time
论文作者
论文摘要
我们考虑解决随机平价游戏的问题。我们证明，平价游戏会超过$ d_p $以上的相位过渡阈值，因此，当定义游戏的图度具有$ d> d_p $的图表时，就会存在一种多项式时间算法，当节点数量到无限时，可以以很高的可能性解决游戏。我们进一步提出了SWCP（自赢的周期传播）算法，并表明，当该学位足够大时，SWCP的可能性很高。此外，SWCP的复杂性是多项式$ o \ big（| {\ cal V} |^2 + | {\ cal V} || {\ cal e} | \ big）$。 SWCP的设计基于玩家各自的子图中特定类型循环的外观的阈值。我们进一步表明，可以在$ o（| {\ cal v} |）$的时间内解决非SPARSE游戏，并具有很高的概率，并就$ d = 2 $ case的硬度发出了猜想。
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.
