Stable gonality is computable

Stable gonality is a multigraph parameter that measures the complexity of a graph. It is defined using maps to trees. Those maps, in some sense, divide the edges equally over the edges of the tree; stable gonality asks for the map with the minimum number of edges mapped to each edge of the tree. This parameter is related to treewidth, but unlike treewidth, it distinguishes multigraphs from their underlying simple graphs. Stable gonality is relevant for problems in number theory. In this paper, we show that deciding whether the stable gonality of a given graph is at most a given integer $k$ belongs to the class NP, and we give an algorithm that computes the stable gonality of a graph in $O((1.33n)^nm^m \text{poly}(n,m))$ time.Comment: 15 pages; v2 minor changes; v3 minor change

Computing graph gonality is hard

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.Comment: The previous version only dealt with hardness of the divisorial gonality. The current version also shows hardness of stable gonality and discusses the relation between the two graph parameter

Parameterized Complexity of Scheduling Chains of Jobs with Delays

In this paper, we consider the parameterized complexity of the following scheduling problem. We must schedule a number of jobs on m machines, where each job has unit length, and the graph of precedence constraints consists of a set of chains. Each precedence constraint is labelled with an integer that denotes the exact (or minimum) delay between the jobs. We study different cases; delays can be given in unary and in binary, and the case that we have a single machine is discussed separately. We consider the complexity of this problem parameterized by the number of chains, and by the thickness of the instance, which is the maximum number of chains whose intervals between release date and deadline overlap. We show that this scheduling problem with exact delays in unary is W[t]-hard for all t, when parameterized by the thickness, even when we have a single machine (m = 1). When parameterized by the number of chains, this problem is W[1]-complete when we have a single or a constant number of machines, and W[2]-complete when the number of machines is a variable. The problem with minimum delays, given in unary, parameterized by the number of chains (and as a simple corollary, also when parameterized by the thickness) is W[1]-hard for a single or a constant number of machines, and W[2]-hard when the number of machines is variable. With a dynamic programming algorithm, one can show membership in XP for exact and minimum delays in unary, for any number of machines, when parameterized by thickness or number of chains. For a single machine, with exact delays in binary, parameterized by the number of chains, membership in XP can be shown with branching and solving a system of difference constraints. For all other cases for delays in binary, membership in XP is open

Discrete and metric divisorial gonality can be different

This paper compares the divisorial gonality of a finite graph $G$ to the divisorial gonality of the associated metric graph $\Gamma(G,\mathbb{1})$ with unit lengths. We show that $\text{dgon}(\Gamma(G,\mathbb{1}))$ is equal to the minimal divisorial gonality of all regular subdivisions of $G$, and we provide a class of graphs for which this number is strictly smaller than the divisorial gonality of $G$. This settles a conjecture of M. Baker in the negative.Comment: 15 pages, 4 figures. Changes: improved Lemma 4.4, added Proposition 5.3, changed open question

Discrete and metric divisorial gonality can be different

This paper compares the divisorial gonality of a finite graph G to the divisorial gonality of the associated metric graph Γ(G,1) with unit lengths. We show that dgon(Γ(G,1)) is equal to the minimal divisorial gonality of all regular subdivisions of G, and we provide a class of graphs for which this number is strictly smaller than the divisorial gonality of G. This settles a conjecture of M. Baker [3, Conjecture 3.14] in the negative

Complexity of Graph Problems: Gonality, Colouring and Scheduling

In this thesis we study several graph problems. In the first part, we study the ‘gonality’ of graphs. This is a measure for the complexity of a graph. There are several variants of gonality. The divisorial gonality can be defined as a chip-firing game, the stable gonality is defined using morphims to trees. We show that it is NP-complete to compute the gonality of a graph. Moreover, we show that the stable gonality is computable. In the second part, we study a variant of graph colouring: rainbow vertex colouring. Here, we colour the vertices of a graph, such that every pair of vertices is connected by a rainbow path: a path in which all internal vertices have different colours. It is NP-hard to compute the minimum number of colours that is needed for such a colouring. We study this problem on several graph classes and give polynomial time algorithms. In the last part, we study a scheduling problem. In this problem several chains of jobs are given. Between every two consecutive jobs in a chain there is a specified amount of delay. Besides, every chain has a release date and deadline. The question is whether there exists a schedule that satisfies all constraints. We study several variants of this problem, and consider several parameters, such as the number of chains. All these variants are W[1]-hard, and some are even W[t]-hard for all t

Complexity of Graph Problems: Gonality, Colouring and Scheduling

In this thesis we study several graph problems. In the first part, we study the ‘gonality’ of graphs. This is a measure for the complexity of a graph. There are several variants of gonality. The divisorial gonality can be defined as a chip-firing game, the stable gonality is defined using morphims to trees. We show that it is NP-complete to compute the gonality of a graph. Moreover, we show that the stable gonality is computable. In the second part, we study a variant of graph colouring: rainbow vertex colouring. Here, we colour the vertices of a graph, such that every pair of vertices is connected by a rainbow path: a path in which all internal vertices have different colours. It is NP-hard to compute the minimum number of colours that is needed for such a colouring. We study this problem on several graph classes and give polynomial time algorithms. In the last part, we study a scheduling problem. In this problem several chains of jobs are given. Between every two consecutive jobs in a chain there is a specified amount of delay. Besides, every chain has a release date and deadline. The question is whether there exists a schedule that satisfies all constraints. We study several variants of this problem, and consider several parameters, such as the number of chains. All these variants are W[1]-hard, and some are even W[t]-hard for all t

Stable gonality is computable

Stable gonality is a multigraph parameter that measures the complexity of a graph. It is defined using maps to trees. Those maps, in some sense, divide the edges equally over the edges of the tree; stable gonality asks for the map with the minimum number of edges mapped to each edge of the tree. This parameter is related to treewidth, but unlike treewidth, it distinguishes multigraphs from their underlying simple graphs. Stable gonality is relevant for problems in number theory. In this paper, we show that deciding whether the stable gonality of a given graph is at most a given integer $k$ belongs to the class NP, and we give an algorithm that computes the stable gonality of a graph in $O((1.33n)^nm^m \text{poly}(n,m))$ time

Stable divisorial gonality is in NP

Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph G can be defined with help of a chip firing game on G. The stable divisorial gonality of G is the minimum divisorial gonality over all subdivisions of edges of G. In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer k belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof consist of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the number of subdivisions needed for minimum stable divisorial gonality of a graph with n vertices is bounded by $2^{p(n)}$ for a polynomial p