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

TL;DR

This work develops Generalized Rotor Mechanisms (GRM) to unify rotor-routing and abelian sandpiles within a vector-adding framework, enabling a broad analysis of reachability under linear and legal routing across four models. It introduces GRM, defines a linear rotor-routing operator, and provides polynomial-time algorithms for cyclic GRMs while proving $NP$-completeness for legal reachability in the general GRM setting. The results extend classical rotor-routing theory, connect to flow and matching problems via the boundary operator, and establish a rigorous complexity landscape that includes a complete polynomial-time criterion for cyclic instances. The study not only generalizes rotor-routing but also lays groundwork for algebraic treatments (via Smith normal form) and further exploration of the rotor/sandpile duality. These findings have implications for understanding deterministic analogues of random walks and the computational tractability of reachability in complex network routing models.

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.

Figures (18)

• Figure 1: Example of a transition in an abelian sandpile graph. The values represent the number of particles at each vertex.
• Figure 2: A rotor multigraph $G_1$ with no sinks, which is strongly connected. Every vertex has out-degree $3$ and in-degree $3$. As an example, we have $\text{head}(a_{2,0})=v_0$ and $\text{tail}(a_{2,0})=v_2$. The rotor order at every vertex is given by anticlockwise ordering; e.g. $\theta(a_{2,0})=a'_{2,0}$, $\theta(a'_{2,0})=a_{2,1}$ and $\theta(a_{2,1})=a_{2,0}$. A rotor configuration $\rho_1$ with $\rho_1(v_0)=a_{0,1}$, $\rho_1(v_1)=a'_{1,2}$ and $\rho_1(v_2)=a_{2,1}$ is depicted in bold.
• Figure 3: A stopping rotor multigraph $G_2$, with two sinks $s_0$ and $s_1$.
• Figure 4: With the same graph $G_1$ as in Fig \ref{['fig:exempleG1']}: on the left, a rotor configuration $\rho_1$ (arcs in bold) and a particle configuration $\sigma_0$ (numbers in vertices) are given. The rotor walk $(\rho_1,\sigma_1), (\rho_2,\sigma_2), (\rho_3,\sigma_3)$ is legal, and consists of routing a particle in $v_0$ and a particle in $v_1$. The resulting configurations $(\rho_3,\sigma_3)$ are given on the right.
• Figure 5: A graph with $\sigma$ given on the left and $\sigma'$ on the right. There is a single linear routing vector $\alpha$ from $\sigma$ to $\sigma'$ which is $2a_1+a_2$. However, none of the three sequences $(a_1, a_1, a_2)$, $(a_1, a_2, a_1)$, $(a_2, a_1, a_1)$ are legal because the middle vertex remains nonpositive at all times.

Theorems & Definitions (50)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Proposition 7
• Proposition 8
• proof
• Proposition 9