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.
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Bazgan, C., Beaujean, P., Gourdin, É. (2018). Relaxation and Matrix Randomized Rounding for the Maximum Spectral Subgraph Problem. In: Kim, D., Uma, R., Zelikovsky, A. (eds) Combinatorial Optimization and Applications. COCOA 2018. Lecture Notes in Computer Science(), vol 11346. Springer, Cham. https://doi.org/10.1007/978-3-030-04651-4_8
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