Rotor-routing reachability is easy, chip-firing reachability is hard

TL;DR

This paper compares the computational complexity of reachability problems in rotor-routing and chip-firing abelian networks, showing rotor-routing is polynomial-time decidable while chip-firing is computationally hard in general.

Contribution

It proves rotor-routing reachability is decidable in polynomial time and demonstrates chip-firing reachability is hard, implying significant complexity differences between the two models.

Findings

Rotor-routing reachability is polynomial-time decidable.

Chip-firing reachability is hard in the general case, linked to the polynomial hierarchy.

A simple characterization of reachability configurations is provided.

Abstract

Chip-firing and rotor-routing are two well-studied examples of abelian networks. We study the complexity of their respective reachability problems. We show that the rotor-routing reachability problem is decidable in polynomial time, and we give a simple characterization of when a chip-and-rotor configuration is reachable from another one. For chip-firing, it has been known that the reachability problem is in P if we have a class of graphs whose period length is polynomial (for example, Eulerian digraphs). Here we show that in the general case, chip-firing reachability is hard in the sense that if the chip-firing reachability problem were in P for general digraphs, then the polynomial hierarchy would collapse to NP. We encode graphs by their adjacency matrix, and we encode ribbon structures "succinctly", only remembering the number of consecutive parallel edges.