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Relaxation and Matrix Randomized Rounding for the Maximum Spectral Subgraph Problem

  • Cristina Bazgan
  • Paul Beaujean
  • Éric Gourdin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

Abstract

Modifying the topology of a network to mitigate the spread of an epidemic with epidemiological constant \(\lambda \) amounts to the NP-hard problem of finding a partial subgraph with maximum number of edges and spectral radius bounded above by \(\lambda \). A software-defined network (SDN) 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 \(O(\log n)\) approximation in the case of finding a subgraph with spectral radius bounded by \(\lambda \in (\log n, \lambda _1(G))\) where \(\lambda _1(G)\) is the spectral radius of the input graph and n its number of nodes. We combine this algorithm with a maximum matching algorithm to obtain a \(O(\log ^2 n)\) approximation algorithm for all values of \(\lambda \). 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.

Keywords

Approximation algorithm Relaxation and rounding Semidefinite programming Spectral graph theory Random graphs 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Cristina Bazgan
    • 1
  • Paul Beaujean
    • 1
    • 2
  • Éric Gourdin
    • 2
  1. 1.Université Paris-Dauphine, Université PSL, CNRS, LAMSADEParisFrance
  2. 2.Orange LabsChâtillonFrance

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