# An integer programming framework for critical elements detection in graphs

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## Abstract

This study presents an integer programming framework for minimizing the connectivity and cohesiveness properties of a given graph by removing nodes and edges subject to a joint budgetary constraint. The connectivity and cohesiveness metrics are assumed to be general functions of sizes of the remaining connected components and node degrees, respectively. We demonstrate that our approach encompasses, as special cases (possibly, under some mild conditions), several other models existing in the literature, including minimization of the total number of connected node pairs, minimization of the largest connected component size, and maximization of the number of connected components. We discuss computational complexity issues, derive linear mixed integer programming (MIP) formulations, and describe additional modeling enhancements aimed at improving the performance of MIP solvers. We also conduct extensive computational experiments with real-life and randomly generated network instances under various settings that reveal interesting insights and demonstrate advantages and limitations of the proposed framework.

## Keywords

Critical node detection Critical edge detection Network interdiction Mixed integer programming## Notes

### Acknowledgments

This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. The research of the first author was performed while he held a National Research Council Research Associateship Award at AFRL. The research of the second author was supported by US AFOSR grant FA9550-11-1-0037 and US Air Force Summer Faculty Fellowship. The authors would like to thank the Associate Editor and the anonymous referees for their constructive comments. In addition, the authors are grateful to Gabriel L. Zenarosa, Ruichen Sun, Dmytro Matsypura and Serdar Karademir for their valuable comments while preparing this manuscript.

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