An approximation algorithm for the maximum spectral subgraph problem

• Published in: Journal of combinatorial optimization Vol. 44; no. 3; pp. 1880 - 1899
• Main Authors: Bazgan, Cristina, Beaujean, Paul, Gourdin, Éric
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-10-2022; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Approximation; Combinatorics; Computer Science; Convex and Discrete Geometry; Cybersecurity; Graph theory; Graphs; Malware; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematical programming; Mathematics; Mathematics and Statistics; Network topologies; Operations Research/Decision Theory; Optimization; Reconfiguration; Rounding; Software-defined networking; Theory of Computation; Relaxation and rounding; Semidefinite programming; Approximation algorithm; Random graphs; Spectral graph theory
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-020-00552-w

Modifying the topology of a network to mitigate the spread of an epidemic with epidemiological constant
λ
amounts to the NP-hard problem of finding a partial subgraph with maximum number of edges and spectral radius bounded above by
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
)
approximation in the case of finding a subgraph with spectral radius bounded by
λ
∈
[
log
n
,
λ
1
(
G
)
)
where
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
)
-approximation algorithm for all values of
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.