Chip-firing games on Eulerian digraphs and NP-hardness of computing the rank of a divisor on a graph

TL;DR

This paper proves that determining the rank of a divisor in graph theory, related to chip-firing games, is NP-hard, highlighting computational complexity in graph-theoretic Riemann-Roch analogues.

Contribution

It establishes the NP-hardness of computing the divisor rank on graphs by connecting chip-firing games on Eulerian digraphs to computational complexity.

Findings

Computing divisor rank is NP-hard.

Chip-firing games relate to divisor rank.

Complexity results extend to directed and undirected graphs.

Abstract

Baker and Norine introduced a graph-theoretic analogue of the Riemann-Roch theory. A central notion in this theory is the rank of a divisor. In this paper we prove that computing the rank of a divisor on a graph is NP-hard. The determination of the rank of a divisor can be translated to a question about a chip-firing game on the same underlying graph. We prove the NP-hardness of this question by relating chip-firing on directed and undirected graphs.