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

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