Linear Extensions of Rotor-Routing in Directed Graphs: Reachability Problems

Abstract

We develop a unified framework for rotor-routing that extends the classical model to a broad class of multigraphs equipped with Generalized Rotor Mechanisms (GRM). This perspective places rotor-routing on the same footing as abelian sandpiles by interpreting both as conservative instances of Vector Addition Systems (VAS). Within this framework, routing becomes a linear transformation governed by arc mechanisms, while legality is enforced through non-negativity constraints. We introduce four routing models -- free routing, standard rotor-routing, cyclic GRM routing, and fully general GRM routing -- and study their reachability problems in both the linear and legal settings. Our results generalize previous characterizations for standard rotor-routing and extend them to the GRM setting. In particular, we show that legal reachability in GRM multigraphs is NP-complete, whereas the cyclic GRM routing model, which includes the classical rotor-router, admits a polynomial-time algorithm.