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The general problem in this paper is vertex (node) subset selection with thegoal to contain an infection that spreads in a network. Instead of selectingthe single most important node, this paper deals with the problem of selectingmultiple nodes for removal. As compared to previous work on multiple-nodeselection, the trade-off between cost and benefit is considered. The benefit ismeasured in terms of increasing the epidemic threshold which is a measure ofhow difficult it is for an infection to spread in a network. The cost ismeasured in terms of the number and size of nodes to be removed or controlled.Already in its single-objective instance with a fixed number of $k$ nodes to beremoved, the multiple vertex immunisation problems have been proven to beNP-hard. Several heuristics have been developed to approximate the problem. Inthis work, we compare meta-heuristic techniques with exact methods on theShield-value, which is a sub-modular proxy for the maximal eigenvalue and usedin the current state-of-the-art greedy node-removal strategies. We generaliseit to the multi-objective case and replace the greedy algorithm by a quadraticprogram (QP), which then can be solved with exact QP solvers. The maincontribution of this paper is the insight that, if time permits, exact andproblem-specific methods approximation should be used, which are often farbetter than Pareto front approximations obtained by general meta-heuristics.Based on these, it will be more effective to develop strategies for controllingreal-world networks when the goal is to prevent or contain epidemic outbreaks.This paper is supported by ready to use Python implementation of theoptimization methods and datasets.

Multiple Node Immunisation for Preventing Epidemics on Networks by Exact Multiobjective Optimisation of Cost and Shield-Value