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

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, even for simple graphs. 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.

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