On the complexity of the chip-firing reachability problem

TL;DR

This paper investigates the computational complexity of the chip-firing reachability problem, providing polynomial-time algorithms for specific graph classes and establishing co-NP membership for the general case.

Contribution

It introduces new polynomial-time decision procedures for Eulerian and certain general digraphs, and shows the problem is in co-NP for broader classes.

Findings

Reachability is in P for Eulerian digraphs with multiple edges.

Reachability is in P when the target distribution is recurrent within each strongly connected component.

The problem is in co-NP for general digraphs.

Abstract

In this paper, we study the complexity of the chip-firing reachability problem. We show that for Eulerian digraphs, the reachability problem can be decided in strongly polynomial time, even if the digraph has multiple edges. We also show a special case when the reachability problem can be decided in polynomial time for general digraphs: if the target distribution is recurrent restricted to each strongly connected component. As a further positive result, we show that the chip-firing reachability problem is in co-NP for general digraphs. We also show that the chip-firing halting problem is in co-NP for Eulerian digraphs.