树型多重图上ARRIVAL的多项式时间算法
Polynomial Time Algorithm for ARRIVAL on Tree-like Multigraphs
摘要：
有向图中的转子行走可以被认为是马尔可夫链的确定性版本，其中卵石遵循简单的规则从一个顶点移动到另一个顶点，直到到达终端顶点或汇点。到达问题，如Dohrau等人所定义的，在于确定将到达哪个水槽。虽然行走本身可以采取指数数量的步骤，但该问题属于复杂性类NP $\cap$ co-NP，而不知道是在P中。已经研究了几种变体，其中我们向模型添加一个或两个参与者，定义随机模型的确定性类似物（例如，马尔可夫决策过程，随机游戏）与转子路由规则，而不是随机过渡。相应的决策问题解决的存在的策略，确保一些条件的球员达到汇。已知这些问题是$NP$ -对于一个参与者是完全的，而$PSPACE$ -对于两个参与者是完全的。在这项工作中，我们定义了一类有向图，即树型多重图，这是一个无向树的全局形状的多重图。我们证明了零，一个或两个球员的可达性问题的不同变体可以在线性时间内解决，而转子行走的步骤数仍然可以是指数。为了实现这一点，我们定义了一个返回流的概念，它计算卵石在图的子树中反弹的次数。
英文摘要：
A rotor walk in a directed graph can be thought of as a deterministic version of a Markov Chain, where a pebble moves from vertex to vertex following a simple rule until a terminal vertex, or sink, is reached. The ARRIVAL problem, as defined by Dohrau and al., consists in determining which sink will be reached. While the walk itself can take an exponential number of steps, this problem belongs to the complexity class NP$\cap$co-NP without being known to be in P. Several variants have been studied where we add one or two players to the model, defining deterministic analogs of stochastic models (e.g., Markovian decision processes, Stochastic Games) with rotor-routing rules instead of random transitions. The corresponding decision problem address the existence of strategies for players that ensure some condition on the reached sink. These problems are known to be $NP$-complete for one player and $PSPACE$-complete for two players. In this work, we define a class of directed graphs, namely tree-like multigraphs, which are multigraphs having the global shape of an undirected tree. We prove that the different variants of the reachability problem with zero, one, or two players can be solved in linear time, while the number of steps of rotor walks can still be exponential. To achieve this, we define a notion of return flow, which counts the number of times the pebble will bounce back in subtrees of the graph.
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