AI 中文总结
本文针对Chechik等人2016年声称的DGG线性时间算法给出反例，并建立有向s-t BP到DGG的线性时间归约，关联了两个公开问题的可解性。
AI 中文摘要
Chechik、Kaplan、Thorup、Zamir和Zwick（STACS 2016）声称提出了一种简单的确定性线性时间比较-based算法，用于求解确定性双人、回合制、零和终端支付博弈，也称为确定性图博弈（DGGs）。我们针对他们的算法给出了一个反例。我们还给出了从有向s-t瓶颈路径（BP）问题到DGG问题的确定性线性时间归约。因此，若存在计算DGG中指定起始顶点值的线性时间比较-based算法，将得到有向s-t BP问题的线性时间比较-based算法。自Gabow和Tarjan给出其O(m log* n)时间算法以来，有向s-t BP问题是否存在此类算法一直是公开问题。因此，DGG公开问题的正面解决也将解决对应的有向s-t BP公开问题。
英文摘要
Chechik, Kaplan, Thorup, Zamir, and Zwick (STACS 2016) claimed a simple deterministic linear-time comparison-based algorithm for solving deterministic two-player, turn-based, zero-sum terminal-payoff games, also known as deterministic graphical games (DGGs). We give a counterexample to their algorithm. We also give a deterministic linear-time reduction from the directed $s$-$t$ bottleneck path (BP) problem to the DGG problem. Consequently, a linear-time comparison-based algorithm for computing the value of a designated start vertex in a DGG would yield a linear-time comparison-based algorithm for directed $s$-$t$ BP. Whether directed $s$-$t$ BP admits such an algorithm has remained open since Gabow and Tarjan gave their $\mathcal{O}(m\log^* n)$-time algorithm. Thus, a positive resolution of the open question for DGGs would also resolve the corresponding open question for directed $s$-$t$ BP.
