ARRIVAL：CLS的下一站
ARRIVAL: Next Stop in CLS
摘要：
我们研究了到达的计算复杂性，一个零玩家游戏的$n$ -顶点开关图介绍了Dohrau，Gärtner，Kohler，Matoušek和Welzl。他们表明，决定终止这个游戏的问题包含在$\text{NP} \cap \text{coNP}$。卡提克角S.最近介绍了一个搜索变体的ARRIVAL，并表明它是在复杂性类PLS。在这项工作中，我们显着提高了已知的上界的决策和搜索变种的ARRIVAL。首先，我们解决了Dohrau等人提出的一个问题，并证明了ARRIVAL的决策变量在$\text{UP} \cap \text{coUP}$中。其次，我们证明了搜索变体ARRIVAL包含在CLS中。第三，我们给出了一个随机的$\mathcal{O}(1.4143^n)$时间算法来解决这两个变量。我们的主要技术贡献是（a）一个有效的可验证的表征的唯一证人终止的到达游戏，和（b）一个有效的方式从状态空间的游戏采样。我们发现，找到唯一的证人的问题是包含在CLS中，而它以前被证明是FPSPACE完全的。有效的采样过程产生的第一个算法的问题，预期运行时$\mathcal{O}(c^n)$与$c<2$。
英文摘要：
We study the computational complexity of ARRIVAL, a zero-player game on $n$-vertex switch graphs introduced by Dohrau, Gärtner, Kohler, Matoušek, and Welzl. They showed that the problem of deciding termination of this game is contained in $\text{NP} \cap \text{coNP}$. Karthik C. S. recently introduced a search variant of ARRIVAL and showed that it is in the complexity class PLS. In this work, we significantly improve the known upper bounds for both the decision and the search variants of ARRIVAL. First, we resolve a question suggested by Dohrau et al. and show that the decision variant of ARRIVAL is in $\text{UP} \cap \text{coUP}$. Second, we prove that the search variant of ARRIVAL is contained in CLS. Third, we give a randomized $\mathcal{O}(1.4143^n)$-time algorithm to solve both variants. Our main technical contributions are (a) an efficiently verifiable characterization of the unique witness for termination of the ARRIVAL game, and (b) an efficient way of sampling from the state space of the game. We show that the problem of finding the unique witness is contained in CLS, whereas it was previously conjectured to be FPSPACE-complete. The efficient sampling procedure yields the first algorithm for the problem that has expected runtime $\mathcal{O}(c^n)$ with $c<2$.
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