A Row Generation Algorithm for Finding Optimal Burning Sequences of Large Graphs

Authors Felipe de Carvalho Pereira , Pedro Jussieu de Rezende , Tallys Yunes , Luiz Fernando Batista Morato



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Author Details

Felipe de Carvalho Pereira
  • Institute of Computing, University of Campinas, Brazil
Pedro Jussieu de Rezende
  • Institute of Computing, University of Campinas, Brazil
Tallys Yunes
  • Miami Herbert Business School, University of Miami, Coral Gables, FL, USA
Luiz Fernando Batista Morato
  • Institute of Computing, University of Campinas, Brazil

Acknowledgements

We sincerely thank the anonymous referees for their valuable feedback.

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Felipe de Carvalho Pereira, Pedro Jussieu de Rezende, Tallys Yunes, and Luiz Fernando Batista Morato. A Row Generation Algorithm for Finding Optimal Burning Sequences of Large Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 94:1-94:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.94

Abstract

We propose an exact algorithm for the Graph Burning Problem (GBP), an NP-hard optimization problem that models the spread of influence on social networks. Given a graph G with vertex set V, the objective is to find a sequence of k vertices in V, namely, v₁, v₂, … , v_k, such that k is minimum and ⋃_{i=1}^{k} {u∈V: d(u,v_i) ≤ k-i} = V, where d(u,v) denotes the distance between u and v. We formulate the problem as a set covering integer programming model and design a row generation algorithm for the GBP. Our method exploits the fact that a very small number of covering constraints is often sufficient for solving the integer model, allowing the corresponding rows to be generated on demand. To date, the most efficient exact algorithm for the GBP, denoted here by GDCA, is able to obtain optimal solutions for graphs with up to 14,000 vertices within two hours of execution. In comparison, our algorithm finds provably optimal solutions approximately 236 times faster, on average, than GDCA. For larger graphs, memory space becomes a limiting factor for GDCA. Our algorithm, however, solves real-world instances with more than 3 million vertices in less than 19 minutes, increasing the size of graphs for which optimal solutions are known by a factor of 200. Additionally, we conduct tests on the proposed algorithm using a series of challenging instances composed of grid graphs containing up to 5,000 vertices. As a result, we achieve novel optimal solutions and tight optimality gaps that have not been previously reported in the literature.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Combinatorial optimization
  • Mathematics of computing → Integer programming
Keywords
  • Graph Burning
  • Burning Number
  • Burning Sequence
  • Set Covering
  • Integer Programming
  • Row Generation

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