APX-Hardness and Approximation for the k-Burning Number Problem

Mondal, Debajyoti; Parthiban, N.; Kavitha, V.; Rajasingh, Indra

doi:10.1007/978-3-030-68211-8_22

Cited by 5 publications

(8 citation statements)

References 20 publications

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“…García-Díaz et al [10] have given a (3 − 2/b)-approximation algorithm where b is the burning number of the input graph. Mondal et al [17] have shown the graph burning problem to be APX-hard, even in a generalized setting where k = O(1) vertices can be chosen to initiate the fire at each step. They gave a 3-approximation algorithm for this generalized version [17].…”

Section: Related Resultsmentioning

confidence: 99%

“…Mondal et al [17] have shown the graph burning problem to be APX-hard, even in a generalized setting where k = O(1) vertices can be chosen to initiate the fire at each step. They gave a 3-approximation algorithm for this generalized version [17]. Since the introduction of the graph burning problem [5], a rich body of literature examines the upper and lower bound on the graph burning number for various classes of graphs [20,14,8] as well as the parameterized complexity of computing the burning number [13].…”

Section: Related Resultsmentioning

confidence: 99%

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Burning Number for the Points in the Plane

Keil¹,

Mondal²,

Moradi³

2022

Preprint

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The burning process on a graph G starts with a single burnt vertex, and at each subsequent step, burns the neighbors of the currently burnt vertices, as well as one other unburnt vertex. The burning number of G is the smallest number of steps required to burn all the vertices of the graph. In this paper, we examine the problem of computing the burning number in a geometric setting. The input is a set of points P in the Euclidean plane. The burning process starts with a single burnt point, and at each subsequent step, burns all the points that are within a distance of one unit from the currently burnt points and one other unburnt point. The burning number of P is the smallest number of steps required to burn all the points of P . We call this variant point burning. We consider another variant called anywhere burning, where we are allowed to burn any point of the plane. We show that point burning and anywhere burning problems are both NP-complete, but (2 + ε) approximable for every ε > 0. Moreover, if we put a restriction on the number of burning sources that can be used, then the anywhere burning problem becomes NP-hard to approximate within a factor of 2 √ 3 − ε.

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Section: Related Resultsmentioning

confidence: 99%

Section: Related Resultsmentioning

confidence: 99%

Burning Number for the Points in the Plane

Keil¹,

Mondal²,

Moradi³

2022

Preprint

Self Cite

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“…A graph decision problem is APX-hard if there is a polynomial time approximation scheme reduction from every problem in APX to that problem. In [33], it was proven that the Graph Burning problem is APX-hard, answering a question from [8]. It was also proven in [33] that even if the burning sources are given as an input, computing a burning sequence itself is NP-hard.…”

Section: Complexity Of Graph Burningmentioning

confidence: 99%

“…In [33], it was proven that the Graph Burning problem is APX-hard, answering a question from [8]. It was also proven in [33] that even if the burning sources are given as an input, computing a burning sequence itself is NP-hard.…”

Section: Complexity Of Graph Burningmentioning

confidence: 99%

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A survey of graph burning

Bonato¹

2020

Preprint

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Graph burning is a deterministic, discrete-time process that models how influence or contagion spreads in a graph. Associated to each graph is its burning number, which is a parameter that quantifies how quickly the influence spreads. We survey results on graph burning, focusing on bounds, conjectures, and algorithms related to the burning number. We will discuss state-of-the-art results on the burning number conjecture, burning numbers of graph classes, and algorithmic complexity. We include a list of conjectures, variants, and open problems on graph burning.

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Improved and Generalized Algorithms for Burning a Planar Point Set

Gokhale

Mark

Mondal

2023

WALCOM: Algorithms and Computation

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Given a set P of points in the plane, a point burning process is a discrete time process to burn all the points of P where fires must be initiated at the given points. Specifically, the point burning process starts with a single burnt point from P , and at each subsequent step, burns all the points in the plane that are within one unit distance from the currently burnt points, as well as one other unburnt point of P (if exists). The point burning number of P is the smallest number of steps required to burn all the points of P . If we allow the fire to be initiated anywhere, then the burning process is called an anywhere burning process, and the corresponding burning number is called anywhere burning number. Computing the point and anywhere burning number is known to be NP-hard. In this paper we show that both these problems admit PTAS in one dimension. We then show that in two dimensions, point burning and anywhere burning are (1.96296 + ε) and (1.92188 + ε) approximable, respectively, for every ε > 0, which improves the previously known (2 + ε) factor for these problems. We also observe that a known result on set cover problem can be leveraged to obtain a 2-approximation for burning the maximum number of points in a given number of steps. We show how the results generalize if we allow the points to have different fire spreading rates. Finally, we prove that even if the burning sources are given as input, finding a point burning sequence itself is NP-hard.

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APX-Hardness and Approximation for the k-Burning Number Problem

Cited by 5 publications

References 20 publications

Burning Number for the Points in the Plane

Burning Number for the Points in the Plane

A survey of graph burning

Improved and Generalized Algorithms for Burning a Planar Point Set

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