# An integer programming framework for critical elements detection in graphs

- 769 Downloads
- 44 Citations

## 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.

## References

- Addis B, Di Summa M, Grosso A (2013) Identifying critical nodes in undirected graphs: Complexity results and polynomial algorithms for the case of bounded treewidth. Discrete Applied Mathematics 161(16/17):2349–2360CrossRefzbMATHMathSciNetGoogle Scholar
- Arulselvan A, Commander CW, Elefteriadou L, Pardalos PM (2009) Detecting critical nodes in sparse graphs. Computers & Operations Research 36(7):2193–2200CrossRefzbMATHMathSciNetGoogle Scholar
- Arulselvan A, Commander CW, Pardalos PM, Shylo O (2007) Managing network risk via critical node identification, Risk Management in Telecommunication Networks. Springer, HeidelbergGoogle Scholar
- Arulselvan A, Commander CW, Shylo O, Pardalos PM (2011) Cardinality-constrained critical node detection problem. In: Gulpinar N, Harrison P, Rustem B (eds) Performance Models and Risk Management in Communications Systems, Springer Optimization and Its Applications, vol 46. Springer, New York, pp 79–91Google Scholar
- Bodlaender HL, Hendriks A, Grigoriev A, Grigorieva NV (2010) The valve location problem in simple network topologies. INFORMS Journal on Computing 22(3):433–442CrossRefzbMATHMathSciNetGoogle Scholar
- Borgatti SP (2006) Identifying sets of key players in a social network. Computational & Mathematical Organization Theory 12(1):21–34CrossRefzbMATHGoogle Scholar
- Chung F, Lu L (2002) Connected components in random graphs with given expected degree sequences. Annals of Combinatorics 6(2):125–145CrossRefzbMATHMathSciNetGoogle Scholar
- Chung F, Lu L (2006) The volume of the giant component of a random graph with given expected degrees. SIAM Journal on Discrete Mathematics 20(2):395–411CrossRefzbMATHMathSciNetGoogle Scholar
- COLOR02/03/04: Graph Coloring and its Generalizations. http://mat.gsia.cmu.edu/COLOR03/. Last accessed September 9, 2013
- Davis TA, Hu Y (2011) The university of florida sparse matrix collection. ACM Transactions on Mathematical Software 38(1):1–25MathSciNetGoogle Scholar
- Di Summa M, Grosso A, Locatelli M (2012) Branch and cut algorithms for detecting critical nodes in undirected graphs. Computational Optimization and Applications 53(3):649–680CrossRefzbMATHMathSciNetGoogle Scholar
- DIMACS. 10th DIMACS Implementation Challenge. Available at http://www.cc.gatech.edu/dimacs10/index.shtml, last accessed September 9, 2013, 2011
- Dinh TN, Xuan Y, Thai MT, Pardalos PM, Znati T (2012) On new approaches of assessing network vulnerability: hardness and approximation. IEEE/ACM Transactions on Networking 20(2):609–619CrossRefGoogle Scholar
- Dinh T.N., Xuan Y., Thai M.T., Park E.K., Znati T. On approximation of new optimization methods for assessing network vulnerability. In INFOCOM, 2010 Proceedings IEEE, pages 1–9, March 2010.Google Scholar
- Erdős P, Rényi A (1959) On random graphs. Publicationes Mathematicae Debrecen 6:290–297MathSciNetGoogle Scholar
- Faloutsos M., Faloutsos P., Faloutsos C. (1999 ) On power-law relationships of the internet topology. In Proceedings of the ACM-SIGCOMM Conference on Applications, Technologies, Architectures, and Protocols for Computer, Communication, pp. 251–262Google Scholar
- FICO™ Xpress Optimization Suite 7.5. http://www.fico.com Last accessed September 9, 2013
- Garey M, Johnson D (1979) Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman and Co., New YorkzbMATHGoogle Scholar
- Hewett R. (2011) Toward identification of key breakers in social cyber-physical networks. In Systems, Man, and Cybernetics (SMC), 2011 IEEE International Conference on, pages 2731–2736,Google Scholar
- Houck DJ, Kim E, O’Reilly GP, Picklesimer DD, Uzunalioglu H (2004) A network survivability model for critical national infrastructures. Bell Labs Technical Journal 8(4):153–172CrossRefGoogle Scholar
- Köppe Matthias, Louveaux Quentin, Weismantel Robert (2008) Intermediate integer programming representations using value disjunctions. Discrete Optimization 5(2):293–313CrossRefzbMATHMathSciNetGoogle Scholar
- Krebs V. . Uncloaking terrorist networks. First Monday, 7(4), 2002. Available at http://journals.uic.edu/ojs/index.php/fm/article/view/941, last accessed September 9, 2013
- Lusseau D, Schneider K, Boisseau OJ, Haase P, Slooten E, Dawson SM (2003) The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations. Behavioral Ecology and Sociobiology 54(4):396–405CrossRefGoogle Scholar
- Matisziw TC, Murray AT (2009) Modeling \(s-t\) path availability to support disaster vulnerability assessment of network infrastructure. Computers & Operations Research 36(1):16–26CrossRefzbMATHGoogle Scholar
- Myung Y-S, Kim H-J (2004) A cutting plane algorithm for computing k-edge survivability of a network. European Journal of Operational Research 156(3):579–589CrossRefzbMATHMathSciNetGoogle Scholar
- Newman MEJ (2003) The structure and function of complex networks. SIAM Review 45:167–256CrossRefzbMATHMathSciNetGoogle Scholar
- Oosten M, Rutten JHGC, Spieksma FCR (2007) Disconnecting graphs by removing vertices: a polyhedral approach. Statistica Neerlandica 61(1):35–60CrossRefzbMATHMathSciNetGoogle Scholar
- Ortiz-Arroyo D., Hussain D.M. (2008) An information theory approach to identify sets of key players. In Proceedings of the 1st European Conference on Intelligence and Security Informatics, EuroISI ’08, pages 15–26, Berlin, Heidelberg, Springer-Verlag.Google Scholar
- Power Systems Test Case Archive, 118 Bus Power Flow Test Case. Available at http://www.ee.washington.edu/research/pstca/pf118/pg_tca118bus.htm, last accessed September 9, 2013
- Reka A, Barabási A-L (2002) Statistical mechanics of complex networks. Reviews of Modern Physics 74:47–97CrossRefMathSciNetGoogle Scholar
- Resende MGC, Pardalos PM (eds) (2006) Handbook of optimization in telecommunications. Springer,Google Scholar
- Shen S, Smith JC (2012) Polynomial-time algorithms for solving a class of critical node problems on trees and series-parallel graphs. Networks 60(2):103–119zbMATHMathSciNetGoogle Scholar
- Shen S, Smith JC, Goli R (2012a) Exact interdiction models and algorithms for disconnecting networks via node deletions. Discrete Optimization 9(3):172–188Google Scholar
- Shen Y., Nguyen N.P., Xuan Y., Thai M.T. (2012b) On the Discovery of Critical Links and Nodes for Assessing Network Vulnerability. IEEE Transactions on Networking, to appear,Google Scholar
- Ventresca M (2012) Global search algorithms using a combinatorial unranking-based problem representation for the critical node detection problem. Computers & Operations Research 39(11):2763–2775CrossRefzbMATHMathSciNetGoogle Scholar
- Ventresca M, Aleman D (2014) A derandomized approximation algorithm for the critical node detection problem. Computers & Operations Research 43:261–270CrossRefMathSciNetGoogle Scholar
- Veremyev A., Boginski V., Pasiliao E.L. (2013) Exact identification of critical nodes in sparse networks via new compact formulations. Optimization Letters, pages 1–15Google Scholar
- Walteros JL, Pardalos PM (2012) Selected topics in critical element detection. In: Daras Nicholas J (ed) Applications of Mathematics and Informatics in Military Science, Springer Optimization and Its Applications, vol 71. Springer, New York, pp 9–26CrossRefGoogle Scholar
- Yannakakis M. Node-and edge-deletion np-complete problems. In Proceedings of the tenth annual ACM symposium on Theory of computing, STOC ’78, pages 253–264, New York, NY, USA, 1978. ACM.Google Scholar
- Zachary WW (1977) An information flow model for conflict and fission in small groups. Journal of Anthropological Research 33:452–473Google Scholar
- Zwaan R, Berger A, Grigoriev A (2011) How to cut a graph into many pieces. In: Ogihara Mitsunori, Tarui Jun (eds) Theory and Applications of Models of Computation, Lecture Notes in Computer Science, vol 6648. Springer, Berlin Heidelberg, pp 184–194CrossRefGoogle Scholar