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An approximation algorithm for the maximum spectral subgraph problem

Author

Listed:
  • Cristina Bazgan

    (Université Paris-Dauphine, Université PSL, CNRS, LAMSADE)

  • Paul Beaujean

    (Université Paris-Dauphine, Université PSL, CNRS, LAMSADE
    Orange Labs)

  • Éric Gourdin

    (Orange Labs)

Abstract

Modifying the topology of a network to mitigate the spread of an epidemic with epidemiological constant $$\lambda $$ λ amounts to the NP-hard problem of finding a partial subgraph with maximum number of edges and spectral radius bounded above by $$\lambda $$ λ . A software-defined network capable of real-time topology reconfiguration can then use an algorithm for finding such subgraph to quickly remove spreading malware threats without deploying specific security countermeasures. In this paper, we propose a novel randomized approximation algorithm based on the relaxation and rounding framework that achieves a $$O(\log n)$$ O ( log n ) approximation in the case of finding a subgraph with spectral radius bounded by $$\lambda \in [\log n, \lambda _1(G))$$ λ ∈ [ log n , λ 1 ( G ) ) where $$\lambda _1(G)$$ λ 1 ( G ) is the spectral radius of the input graph and n is the number of nodes. We combine this algorithm with a maximum matching algorithm to obtain a $$O(\log ^2 n)$$ O ( log 2 n ) -approximation algorithm for all values of $$\lambda $$ λ . We also describe how the mathematical programming formulation we give has several advantages over previous approaches which attempted at finding a subgraph with minimum spectral radius given an edge removal budget. Finally, we show that the analysis of our randomized rounding scheme is essentially tight by relating it to classical results from random graph theory.

Suggested Citation

  • Cristina Bazgan & Paul Beaujean & Éric Gourdin, 2022. "An approximation algorithm for the maximum spectral subgraph problem," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1880-1899, October.
  • Handle: RePEc:spr:jcomop:v:44:y:2022:i:3:d:10.1007_s10878-020-00552-w
    DOI: 10.1007/s10878-020-00552-w
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    References listed on IDEAS

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    1. Jiawang Nie, 2011. "Polynomial Matrix Inequality and Semidefinite Representation," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 398-415, August.
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