Computing graph gonality is hard

Abstract

There are several notions of gonality for graphs. The divisorial gonality dgon(G) of a graph G is the smallest degree of a divisor of positive rank in the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the minimum degree of a finite harmonic morphism from a refinement of G to a tree, as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and sgon(G) are NP-hard by a reduction from the maximum independent set problem and the vertex cover problem, respectively. Both constructions show that computing gonality is moreover APX-hard.

Stable Divisorial Gonality is in NP

“…In this paper, we showed that the problem to decide whether the stable divisorial gonality of a given graph is at most a given number K belongs to the class NP. Together with the NP-hardness result of Gijswijt [10], this shows that the problem is NP-complete. We think our proof technique is interesting: we give a certificate that describes some of the essential aspects of the firing sequences; whether there is a subdivision of the graph for which this certificate describes the firing sequences and thus gives the subdivision that reaches the optimal divisorial gonality can be expressed in an integer linear program.…”

“…As ILP's have certificates with polynomially many bits (see e.g., [12]), and the partial certificate is of polynomial size (see also Lemma 3.4), we have that, using Lemmas 4.1 and 4.3, the problem whether a given graph has divisorial gonality at most a given integer k has a polynomial certificate, which gives our main result. Combined with the NP-hardness by Gijswijt [10], this yields the following theorem.…”

“…It is open whether stable divisorial gonality is in XP. NP-hardness of the notions was shown by Gijswijt [10].…”

Gonality of expander graphs

“…While it is NP-hard to compute the gonality of a graph [Gij15], the complexity arises from the sheer number of divisors that one would be required to check. If we are given a divisor, there is a relatively simple algorithm that determines whether it has positive rank.…”

A New Lower Bound on Graph Gonality

“…Due to its connection to algebraic geometry, this invariant has received a great deal of interest [vDdBG14, Gij15, DJKM16, DJ18, AM19, ADM + 20]. Computing the gonality gon(G) of a graph G is NP-hard [Gij15]. To find an upper bound, one only has to produce an example of a divisor with positive rank, so much of the difficulty comes from finding lower bounds.…”