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Local Treewidth of Random and Noisy Graphs with Applications to Stopping Contagion in Networks

Authors Hermish Mehta, Daniel Reichman

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ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Graph Algorithms
  • Random Graphs
  • Data Structures and Algorithms
  • Discrete Mathematics

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Abstract

We study the notion of local treewidth in sparse random graphs: the maximum treewidth over all k-vertex subgraphs of an n-vertex graph. When k is not too large, we give nearly tight bounds for this local treewidth parameter; we also derive nearly tight bounds for the local treewidth of noisy trees, trees where every non-edge is added independently with small probability. We apply our upper bounds on the local treewidth to obtain fixed parameter tractable algorithms (on random graphs and noisy trees) for edge-removal problems centered around containing a contagious process evolving over a network. In these problems, our main parameter of study is k, the number of initially "infected" vertices in the network. For the random graph models we consider and a certain range of parameters the running time of our algorithms on n-vertex graphs is 2^o(k) poly(n), improving upon the 2^Ω(k) poly(n) performance of the best-known algorithms designed for worst-case instances of these edge deletion problems.

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Hermish Mehta and Daniel Reichman. Local Treewidth of Random and Noisy Graphs with Applications to Stopping Contagion in Networks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.7

Author Details

Hermish Mehta
  • Citadel Securities, Chicago, IL, USA
Daniel Reichman
  • Department of Computer Science, Worcester Polytechnic Institute, MA, USA

Acknowledgements

We are very grateful to Michael Krivelevich who provided numerous valuable comments and links to relevant work. Josh Erde offered useful feedback. We would like to thank the anonymous referees for helpful comments and suggestions. In particular, we thank a reviewer for noting a gap in a claimed proof of a stronger lower bound of Ω(klog d/log n) of the local treewidth of G(n,d/n). An inspiration for this paper has been the operation of the Oncology department at Haddasah Ein Karem hospital during the Covid-19 pandemic. Their professionalism and dedication are greatly acknowledged.

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