Burning a graph is hard

Bessy, Stphane; Bonato, Anthony; Janssen, Jeannette; Rautenbach, Dieter; Roshanbin, Elham

doi:10.1016/j.dam.2017.07.016

Cited by 63 publications

(78 citation statements)

References 12 publications

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“…The burning number of a given graph is the minimum such number; hence, an optimal algorithm burns the graph in a number of rounds that is equal to the burning number. Unfortunately, finding optimal solutions is NP-hard even for elementary graph families [2]. The focus of this paper is to provide approximation algorithms for burning graphs.…”

Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for Graph Burning

Bonato

Kamali

2019

Lecture Notes in Computer Science

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Numerous approaches study the vulnerability of networks against social contagion. Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchronous, discrete rounds. In each round, a fire breaks out at a vertex, and the fire spreads to all vertices that are adjacent to a burning vertex. The selection of vertices where fires start defines a schedule that indicates the number of rounds required to burn all vertices. Given a graph, the objective of an algorithm is to find a schedule that minimizes the number of rounds to burn graph. Finding the optimal schedule is known to be NP-hard, and the problem remains NP-hard when the graph is a tree or a set of disjoint paths. The only known algorithm is an approximation algorithm for disjoint paths, which has an approximation ratio of 1.5.We present approximation algorithms for graph burning. For general graphs, we introduce an algorithm with an approximation ratio of 3. When the graph is a tree, we present another algorithm with approximation ratio 2. Moreover, we consider a setting where the graph is a forest of disjoint paths. In this setting, when the number of paths is constant, we provide an optimal algorithm which runs in polynomial time. When the number of paths is more than a constant, we provide two approximation schemes: first, under a regularity condition where paths have asymptotically equal lengths, we show the problem admits an approximation scheme which is fully polynomial. Second, for a general setting where the regularity condition does not necessarily hold, we provide another approximation scheme which runs in time polynomial in the size of the graph.

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Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for Graph Burning

Bonato

Kamali

2019

Lecture Notes in Computer Science

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“…The formal definition of this problem was provided in the framework of graph theory by Anthony Bonato et al [1]. It was proved that computing a "burning number" is NP-complete [2]. The problem was further studied in papers [3,4], where the bounds for the burning number (i.e., the minimum number of time steps, equal to the minimum size of the set of nodes, which are informed or alarmed "from the outside", not from their neighbor) are analyzed.…”

Section: Introductionmentioning

confidence: 99%

Heuristics for Spreading Alarm throughout a Network

et al. 2019

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This paper provides heuristic methods for obtaining a burning number, which is a graph parameter measuring the speed of the spread of alarm, information, or contagion. For discrete time steps, the heuristics determine which nodes (centers, hubs, vertices, users) should be alarmed (in other words, burned) and in which order, when afterwards each alarmed node alarms its neighbors in the network at the next time step. The goal is to minimize the number of discrete time steps (i.e., time) it takes for the alarm to reach the entire network, so that all the nodes in the networks are alarmed. The burning number is the minimum number of time steps (i.e., number of centers in a time sequence alarmed “from outside”) the process must take. Since the problem is NP complete, its solution for larger networks or graphs has to use heuristics. The heuristics proposed here were tested on a wide range of networks. The complexity of the heuristics ranges in correspondence to the quality of their solution, but all the proposed methods provided a significantly better solution than the competing heuristic.

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“…In this paper, we settle the burning number conjecture for the class of spider graphs, which are trees with exactly one vertex of degree strictly greater than two; see Theorem 7. While spiders might initially appear to be an elementary graph class in the context of graph burning, it was shown in [2] that computing the burning number on spiders is NP-complete. The main ingredient in our proof of the conjecture for spiders relies on new bounds on the burning number of path-forests (that is, disjoint unions of paths); see Lemmas 2 and 3.…”

Section: Introductionmentioning

confidence: 99%

“…The bounds provided in our results improve on the known bound given in [3] of b(G) ≤ ⌈n 1/2 ⌉ + t − 1, where G is a path-forest of order n with t components. As shown in [2], the problem of computing the burning number of path-forests is also NP-complete. Our bounds provide a All graphs we consider are simple, finite, and undirected.…”

Section: Introductionmentioning

confidence: 99%

Bounds on the burning numbers of spiders and path-forests

Bonato

Lidbetter

2019

Theoretical Computer Science

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Graph burning is one model for the spread of memes and contagion in social networks. The corresponding graph parameter is the burning number of a graph G, written b(G), which measures the speed of the social contagion. While it is conjectured that the burning number of a connected graph of order n is at most ⌈ √ n⌉, this remains open in general and in many graph families. We prove the conjectured bound for spider graphs, which are trees with exactly one vertex of degree at least 3. To prove our result for spiders, we develop new bounds on the burning number for path-forests, which in turn leads to a 3 2 -approximation algorithm for computing the burning number of path-forests.1991 Mathematics Subject Classification. 05C05,05C85.

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Burning a graph is hard

Cited by 63 publications

References 12 publications

Approximation Algorithms for Graph Burning

Approximation Algorithms for Graph Burning

Heuristics for Spreading Alarm throughout a Network

Bounds on the burning numbers of spiders and path-forests

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