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Book ChapterDOI

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

TL;DR: In this paper, the k-burning number is defined as the minimum number of steps such that all the vertices can be burned within a given number of rounds, where the last round may have smaller than k unburnt vertices, where all of them are burned.
Abstract: Consider an information diffusion process on a graph G that starts with \(k>0\) burnt vertices, and at each subsequent step, burns the neighbors of the currently burnt vertices, as well as k other unburnt vertices. The k-burning number of G is the minimum number of steps \(b_k(G)\) such that all the vertices can be burned within \(b_k(G)\) steps. Note that the last step may have smaller than k unburnt vertices available, where all of them are burned. The 1-burning number coincides with the well-known burning number problem, which was proposed to model the spread of social contagion. The generalization to k-burning number allows us to examine different worst-case contagion scenarios by varying the spread factor k.
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Journal ArticleDOI
TL;DR: In this paper , the problem is shown to be W[2]-complete parameterized by k and that it does not admit a polynomial kernel parameterised by vertex cover number unless $$\mathrm{NP} \subseteq \mathrm {coNP/poly}$$
Abstract: Abstract Graph Burning asks, given a graph $$G = (V,E)$$ G = ( V , E ) and an integer k , whether there exists $$(b_{0},\dots ,b_{k-1}) \in V^{k}$$ ( b 0 , , b k - 1 ) V k such that every vertex in G has distance at most i from some $$b_{i}$$ b i . This problem is known to be NP-complete even on connected caterpillars of maximum degree 3. We study the parameterized complexity of this problem and answer all questions by Kare and Reddy [IWOCA 2019] about the parameterized complexity of the problem. We show that the problem is W[2]-complete parameterized by k and that it does not admit a polynomial kernel parameterized by vertex cover number unless $$\mathrm {NP} \subseteq \mathrm {coNP/poly}$$ NP coNP / poly . We also show that the problem is fixed-parameter tractable parameterized by clique-width plus the maximum diameter among all connected components. This implies the fixed-parameter tractability parameterized by modular-width, by treedepth, and by distance to cographs. Using a different technique, we show that parameterization by distance to split graphs is also tractable. We finally show that the problem parameterized by max leaf number is XP.

2 citations

References
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Journal ArticleDOI
01 Apr 1941
TL;DR: Let N be a network (or linear graph) such that at each node not more than n lines meet (where n > 2), and no line has both ends at the same node.
Abstract: Let N be a network (or linear graph) such that at each node not more than n lines meet (where n > 2), and no line has both ends at the same node. Suppose also that no connected component of N is an n-simplex. Then it is possible to colour the nodes of N with n colours so that no two nodes of the same colour are joined.

938 citations

Journal ArticleDOI
TL;DR: Four fundamental graph problems, Minimum vertex cover, Maximum independent set, Minimum dominating set and Maximum cut, are shown to be APX-complete even for cubic graphs, unless P = NP, which means these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three.

392 citations

Journal ArticleDOI
TL;DR: A powerful, and yet simple, technique for devising approximation algorithms for a wide variety of NP-complete problems in routing, location, and communication network design is investigated.
Abstract: In this paper a powerful, and yet simple, technique for devising approximation algorithms for a wide variety of NP-complete problems in routing, location, and communication network design is investigated. Each of the algorithms presented here delivers an approximate solution guaranteed to be within a constant factor of the optimal solution. In addition, for several of these problems we can show that unless P = NP, there does not exist a polynomial-time algorithm that has a better performance guarantee.

371 citations

Book ChapterDOI
17 Dec 2014
TL;DR: A new graph parameter called the burning number is introduced, inspired by contact processes on graphs such as graph bootstrap percolation, and graph searching paradigms such as Firefighter, which measures the speed of the spread of contagion in a graph.
Abstract: We introduce a new graph parameter called the burning number, inspired by contact processes on graphs such as graph bootstrap percolation, and graph searching paradigms such as Firefighter. The burning number measures the speed of the spread of contagion in a graph; the lower the burning number, the faster the contagion spreads. We provide a number of properties of the burning number, including characterizations and bounds. The burning number is computed for several graph classes, and is derived for the graphs generated by the Iterated Local Transitivity model for social networks.

64 citations