Computing Higher Graph Gonality is Hard

In the theory of divisors on multigraphs, the r^th divisorial gonality of a graph is the minimum degree of a rank r divisor on that graph. It was proved by Gijswijt et al. that the first divisorial gonality of a finite graph is NP-hard to compute. We generalize their argument to prove that it is NP-hard to compute the r^th divisorial gonality of a finite graph for all r. We use this result to prove that it is NP-hard to compute r^th stable divisorial gonality for a finite graph, and to compute r^th divisorial gonality for a metric graph. We also prove these problems are APX-hard, and we study the NP-completeness of these problems.