简单平均收益博弈中的最快GKK算法
The GKK Algorithm is the Fastest over Simple Mean-Payoff Games
摘要：
研究了Gurvich，Khachyian和Karzanov算法（GKK算法）在不含权为零的简单圈的平均支付博弈上的运行情况。我们提出了一个新的对称分析，降低$O(n^2 N)$上界Pisaruk的迭代次数下降到$N + Ep + Em$，这是小于$nN$，其中$n$是顶点的数量，$N$是最大的绝对值的权重，和$Ep$和$Em$分别是最大的有限能量和双能量值的游戏。由于每次迭代都是在$O(m)$中计算的，因此通过考虑游戏图的结构，这改进了Brim、Chaloupka、Doyen、Gentilini和Raskin的最新伪多项式$O(mnN)$运行时界限。我们补充我们的结果表明，Dorfman，Kaplan和Zwick的分析也适用于GKK算法，因此也受到最先进的组合运行时界的$O(m 2^{n/2})$。
英文摘要：
We study the algorithm of Gurvich, Khachyian and Karzanov (GKK algorithm) when it is ran over mean-payoff games with no simple cycle of weight zero. We propose a new symmetric analysis, lowering the $O(n^2 N)$ upper-bound of Pisaruk on the number of iterations down to $N + Ep + Em$, which is smaller than $nN$, where $n$ is the number of vertices, $N$ is the largest absolute value of a weight, and $Ep$ and $Em$ are respectively the largest finite energy and dual-energy values of the game. Since each iteration is computed in $O(m)$, this improves on the state of the art pseudopolynomial $O(mnN)$ runtime bound of Brim, Chaloupka, Doyen, Gentilini and Raskin, by taking into account the structure of the game graph. We complement our result by showing that the analysis of Dorfman, Kaplan and Zwick also applies to the GKK algorithm, which is thus also subject to the state of the art combinatorial runtime bound of $O(m 2^{n/2})$.
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