Relaxation and Matrix Randomized Rounding for the Maximum Spectral Subgraph Problem

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.