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Burning Two Worlds

Algorithms for Burning Dense and Tree-Like Graphs

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Book cover SOFSEM 2020: Theory and Practice of Computer Science (SOFSEM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12011))

Abstract

Graph burning is a model for the spread of social influence in networks. The objective is to measure how quickly a fire (e.g., a piece of fake news) can be spread in a network. The burning process takes place in discrete rounds. In each round, a new fire breaks out at a selected vertex and burns it. Meanwhile, the old fires extend to their adjacent vertices and burn them. A burning schedule selects where the new fire breaks out in each round, and the burning problem asks for a schedule that burns all vertices in a minimum number of rounds, termed the burning number of the graph. The burning problem is known to be NP-hard even when the graph is a tree or a disjoint set of paths. For connected graphs, it has been conjectured [3] that burning takes at most \(\lceil \sqrt{n} \ \rceil \) rounds.

In this paper, we approach the algorithmic study of graph burning from two directions. First, we consider connected n-vertex graphs with minimum degree \(\delta \). We present an algorithm that burns any such graph in at most \(\sqrt{\frac{24n}{\delta +1}}\) rounds. In particular, for graphs with \(\delta \in \varTheta (n)\), all vertices are burned in a constant number of rounds. More interestingly, even when \(\delta \) is a constant that is independent of n, our algorithm answers the graph-burning conjecture in the affirmative by burning the graph in at most \(\lceil \sqrt{n} \rceil \) rounds. Then, we consider burning connected graphs with bounded pathlength or treelength. This includes many graph families, e.g., interval graphs (pathlength 1) and chordal graphs (treelength 1). We show that any connected graph with pathlength pl and diameter d can be burned in \(\lceil \sqrt{d-1} \rceil + pl\) rounds. Our algorithm ensures an approximation ratio of \(1+o(1)\) for graphs of bounded pathlength. We also give an algorithm that achieves an approximation ratio of \(2+o(1)\) for burning connected graphs of bounded treelength. Our approximation factors are better than the best known approximation factor of 3 for burning general graphs.

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Authors and Affiliations

  1. University of Manitoba, Winnipeg, MB, Canada

    Shahin Kamali, Avery Miller & Kenny Zhang

Authors
  1. Shahin Kamali
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  2. Avery Miller
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  3. Kenny Zhang
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Corresponding author

Correspondence to Avery Miller .

Editor information

Editors and Affiliations

  1. University of Macedonia, Thessaloniki, Greece

    Alexander Chatzigeorgiou

  2. University of Bergamo, Bergamo, Italy

    Riccardo Dondi

  3. Cyprus University of Technology, Limassol, Cyprus

    Herodotos Herodotou

  4. Carnegie Mellon University Qatar, Doha, Qatar

    Christos Kapoutsis

  5. Open University of Cyprus, Nicosia, Cyprus

    Yannis Manolopoulos

  6. University of Cyprus, Nicosia, Cyprus

    George A. Papadopoulos

  7. Paris Dauphine University, Paris, France

    Florian Sikora

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Kamali, S., Miller, A., Zhang, K. (2020). Burning Two Worlds. In: Chatzigeorgiou, A., et al. SOFSEM 2020: Theory and Practice of Computer Science. SOFSEM 2020. Lecture Notes in Computer Science(), vol 12011. Springer, Cham. https://doi.org/10.1007/978-3-030-38919-2_10

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  • DOI: https://doi.org/10.1007/978-3-030-38919-2_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-38918-5

  • Online ISBN: 978-3-030-38919-2

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