Computing graph gonality is hard

Gijswijt, Dion; Smit, Harry; Wegen, Marieke van der

doi:10.1016/j.dam.2020.08.013

Cited by 24 publications

(32 citation statements)

References 18 publications

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“…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 et al [9], 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.…”

Section: Resultssupporting

confidence: 58%

“…Combined with the NP-hardness of STABLE DIVISORIAL GONALITY by Gijswijt et al [9], this yields the following result.…”

Section: Theorem 54 Stable Divisorial Gonality Belongs To the Class Npmentioning

confidence: 69%

“…In this paper, we study the complexity of computing the stable divisorial gonality of graphs: i.e., we look at the complexity of the STABLE DIVISORIAL GONALITY problem: given an undirected graph G = (V , E) and an integer k, decide whether the stable divisorial gonality of G is at most k. It was shown in [9] that divisorial gonality is NP-complete. The same reduction gives that stable divisorial gonality is NP-hard.…”

Section: Introductionmentioning

confidence: 99%

“…there is an algorithm that decides in time O(n f (k) ) whether dgon(G) ≤ k. It is open whether stable divisorial gonality is in XP. NP-hardness of the notions was shown in [9].…”

Section: Introductionmentioning

confidence: 99%

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Stable Divisorial Gonality is in NP

2020

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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 et al., we obtain that stable divisorial gonality is NP-complete. The proof consists of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the total number of subdivisions needed for minimum stable divisorial gonality of a graph with m edges is bounded by mO(mn).

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Section: Resultssupporting

confidence: 58%

“…Combined with the NP-hardness of STABLE DIVISORIAL GONALITY by Gijswijt et al [9], this yields the following result.…”

Section: Theorem 54 Stable Divisorial Gonality Belongs To the Class Npmentioning

confidence: 69%

Section: Introductionmentioning

confidence: 99%

“…there is an algorithm that decides in time O(n f (k) ) whether dgon(G) ≤ k. It is open whether stable divisorial gonality is in XP. NP-hardness of the notions was shown in [9].…”

Section: Introductionmentioning

confidence: 99%

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Stable Divisorial Gonality is in NP

2020

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“…Over the last years, the gonality of metric graphs has been the subject of research in combinatorics and computer science: the gonality of Γ is bounded from below by the treewidth of for any model ( , ℓ) of Γ [vDdBG20], and also-like for Riemann surfaces [YY80]-by an expression involving its volume and the smallest positive eigenvalue of the Laplace operator of Γ [AK16]. Moreover, like treewidth, gonality is NP-complete [Gij15,BvdWvdZ19].…”

Section: Gonality Of Metric Graphsmentioning

confidence: 99%

On the Gonality of Metric Graphs

Draisma¹,

Vargas²

2021

Notices Amer. Math. Soc.

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The last decades have seen an extremely fruitful interplay between Riemann surfaces and graphs with a metric. A deformation process called tropicalisation transforms the former into the latter. Under this process, additional structure on the Riemann surfaces yields additional structure on the metric graphs. For instance, meromorphic functions on Riemann surfaces yield piecewise linear functions on metric graphs. In this manner, theorems in algebraic geometry have deep combinatorial consequences; and conversely, combinatorial arguments can be used to prove theorems in algebraic geometry.

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Problems Hard for Treewidth but Easy for Stable Gonality

Bodlaender

Cornelissen

Wegen

2022

Lecture Notes in Computer Science

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We show that some natural problems that are XNLPhard (hence W[t]-hard for all t) when parameterized by pathwidth or treewidth, become FPT when parameterized by stable gonality, a novel graph parameter based on optimal maps from graphs to trees. The problems we consider are classical flow and orientation problems, such as Undirected Flow with Lower Bounds, Minimum Maximum Outdegree, and capacitated optimization problems such as Capacitated (Red-Blue) Dominating Set. Our hardness claims beat existing results. The FPT algorithms use a new parameter "treebreadth", associated to a weighted tree partition, as well as DP and ILP.

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Computing graph gonality is hard

Cited by 24 publications

References 18 publications

Stable Divisorial Gonality is in NP

Stable Divisorial Gonality is in NP

On the Gonality of Metric Graphs

Problems Hard for Treewidth but Easy for Stable Gonality

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