Algorithmic aspects of rotor-routing and the notion of linear equivalence

TL;DR

This paper introduces a linear equivalence concept for rotor-routing on graphs, enabling efficient computation of related problems and linking rotor-routing dynamics to algebraic graph invariants.

Contribution

It defines linear equivalence for rotor-routing, proves polynomial-time algorithms for related problems, and connects rotor-routing with algebraic graph theory concepts.

Findings

Polynomial-time computability of rotor-routing problems

Equality of rotor-router unicycle-orbits and the Picard group order

Interpretation of Picard group action in terms of linear equivalence

Abstract

We define the analogue of linear equivalence of graph divisors for the rotor-router model, and use it to prove polynomial time computability of some problems related to rotor-routing. Using the connection between linear equivalence for chip-firing and for rotor-routing, we give a simple proof for the fact that the number of rotor-router unicycle-orbits equals the order of the Picard group. We also show that the rotor-router action of the Picard group on the set of spanning in-arborescences can be interpreted in terms of the linear equivalence.