Chip-firing games on Eulerian digraphs and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" display="inline" overflow="scroll"><mml:mstyle mathvariant="bold"><mml:mi>NP</mml:mi></mml:mstyle></mml:math>-hardness of computing the rank of a divisor on a graph

Kiss, Viktor; Tóthmérész, Lilla

doi:10.1016/j.dam.2015.04.030

Cited by 14 publications

(13 citation statements)

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“…Asadi and Backman [3] extend parts of the Baker-Norine theory to directed graphs. Kiss and Tóthmérész [18] show that computing the Baker-Norine rank-a harder problem than deciding whether it is nonnegative-is already NP-hard when L is the Laplacian lattice of a simple undirected graph. )…”

Section: Nonnegative Rankmentioning

confidence: 99%

CoEulerian graphs

Farrell

Levine

2015

Proc. Amer. Math. Soc.

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We suggest a measure of "Eulerianness" of a finite directed graph and define a class of "coEulerian" graphs. These are the graphs whose Laplacian lattice is as large as possible. As an application, we address a question in chip-firing posed by Björner, Lovász, and Shor in 1991, who asked for "a characterization of those digraphs and initial chip configurations that guarantee finite termination." Björner and Lovász gave an exponential time algorithm in 1992. We show that this can be improved to linear time if the graph is coEulerian, and that the problem is NP-complete for general directed multigraphs.

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Section: Nonnegative Rankmentioning

confidence: 99%

CoEulerian graphs

Farrell

Levine

2015

Proc. Amer. Math. Soc.

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“…In Section 3 we give a characterization of non-terminating chip-distributions of Eulerian digraphs using turnback arc sets. This characterization is a variant of some previous results [4,15,13], designed to suit our purposes in the paper. We note that some variant of the specialization of the characterization to undirected graphs plays a central role in each proof of the Riemann-Roch theorem for undirected graphs (see [3,7,17]).…”

Section: Introductionmentioning

confidence: 89%

“…From this it follows that the following quantity, which measures how far a given distribution is from being non-terminating, is well defined. Definition 2.6 ( [13]). For a distribution x ∈ Chip(G), the distance of x from non-terminating distributions is…”

Section: Chip-firingmentioning

confidence: 99%

“…This connection has already turned out to be fruitful in both directions. NP-hardness results in both chip-firing (halting problem of chip-firing games on directed multigraphs [8]) and Riemann-Roch theory (the computation of the rank of a divisor [13]) have recently been proved by translating the problem to the dual language. We note that the graph divisor theory of Baker and Norine is sometimes also referred to as chip-firing, but in this paper by chip-firing we always mean the game of Björner, Lovász and Shor.…”

Section: Introductionmentioning

confidence: 99%

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Chip-firing based methods in the Riemann–Roch theory of directed graphs

Hujter

Tóthmérész

2019

European Journal of Combinatorics

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Baker and Norine proved a Riemann-Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of Björner, Lovász and Shor. We use this connection to prove Riemann-Roch-type results on directed graphs. We give a simple proof for a Riemann-Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann-Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph theoretic notions more explicit via using the chip-firing framework.

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“…The ASM was introduced (in the special case of the two-dimensional square lattice) by the physicists Bak, Tang, and Wiesenfeld [BTW87] as a simple model of self-organized criticality; much of the general, graphical theory was subsequently developed by Dhar [Dha90,Dha99]. The ASM is by now studied in many parts of both physics and pure mathematics: for instance, following the seminal work of Baker and Norine [BN07], it is known that this model is intimately related to tropical algebraic geometry (specifically, divisor theory for tropical curves [GK08,MZ08]); meanwhile, the ASM is studied by probabilists because of its remarkable scaling-limit behavior [PS13,LPS16]; and there are also interesting complexity-theoretic questions related to the ASM, such as, what is the complexity of determining whether a given configuration stabilizes [KT15,FL16]. For more on sandpiles, consult the short survey article [LP10] or the recent textbook [CP18].…”

Section: Introductionmentioning

confidence: 99%

Root system chip-firing I: interval-firing

et al. 2018

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Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confluent from some (but not all) initial configurations of chips. We recast their set-up in terms of root systems: labeled chip-firing can be seen as a root-firing process which allows the moves λ− →λ+α for α ∈ Φ + whenever λ, α ∨ = 0, where Φ + is the set of positive roots of a root system of Type A and λ is a weight of this root system. We are thus motivated to study the exact same root-firing process for an arbitrary root system. Actually, this central root-firing process is the subject of a sequel to this paper. In the present paper, we instead study the interval root-firing processes determined by λ, for any k ≥ 0. We prove that these interval-firing processes are always confluent, from any initial weight. We also show that there is a natural way to consistently label the stable points of these interval-firing processes across all values of k so that the number of weights with given stabilization is a polynomial in k. We conjecture that these Ehrhart-like polynomials have nonnegative integer coefficients.

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Chip-firing games on Eulerian digraphs and NP-hardness of computing the rank of a divisor on a graph

Cited by 14 publications

References 10 publications

CoEulerian graphs

CoEulerian graphs

Chip-firing based methods in the Riemann–Roch theory of directed graphs

Root system chip-firing I: interval-firing

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