Computing higher graph gonality is hard

TL;DR

This paper proves that computing the r-th divisorial gonality of a finite graph, including stable and metric variants, is NP-hard and APX-hard, extending previous results for the first gonality.

Contribution

It generalizes NP-hardness results to all r-th divisorial gonality and related variants, establishing computational complexity for these graph invariants.

Findings

NP-hardness for all r-th divisorial gonality

APX-hardness of the problems

NP-completeness of related decision problems

Abstract

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.