Polynomial Time Algorithm for ARRIVAL on Tree-like Multigraphs

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.