The rank of a divisor on a finite graph: geometry and computation

We study the problem of computing the rank of a divisor on a finite graph, a quantity that arises in the Riemann-Roch theory on the finite graph developed by Baker and Norine (Advances in Mathematics, 215(2): 766-788, 2007). Our work consists of two parts: the first part is an algorithm whose running time is polynomial for a multigraph with a fixed number of vertices. More precisely, our algorithm has running time O(2 )poly(size)(G), where n + 1 is the number of vertices of the graph G. The second part consisits of a new proof of the fact that testing if rank of a divisor is non-negative or not is in the complexity class NP ∩ co−NP and motivated by this proof and its generalisations, we construct a new graph invariant that we call the critical automorphism group of the graph.