Generalized ARRIVAL Problem for Rotor Walks in Path Multigraphs

TL;DR

This paper studies a generalized reachability problem in rotor walks on path multigraphs, revealing its computational complexity and providing algebraic invariants for efficient solutions even in complex scenarios.

Contribution

It introduces a generalized ARRIVAL problem for rotor walks, analyzes its complexity, and develops algebraic invariants based on harmonic functions for efficient resolution.

Findings

The problem is in NP and co-NP for general cases.

Algebraic invariants enable efficient solutions in path multigraphs.

Harmonic functions relate to integer decompositions in rational bases.

Abstract

Rotor walks are cellular automata that determine deterministic traversals of particles in a directed multigraph using simple local rules, yet they can generate complex behaviors. Furthermore, these trajectories exhibit statistical properties similar to random walks. In this study, we investigate a generalized version of the reachability problem known as ARRIVAL in Path Multigraphs, which involves predicting the number of particles that will reach designated target vertices. We show that this problem is in NP and co-NP in the general case. However, we exhibit algebraic invariants for Path Multigraphs that allow us to solve the problem efficiently, even for an exponential configuration of particles. These invariants are based on harmonic functions and are connected to the decomposition of integers in rational bases.