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.
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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):2349–2360
Arulselvan A, Commander CW, Elefteriadou L, Pardalos PM (2009) Detecting critical nodes in sparse graphs. Computers & Operations Research 36(7):2193–2200
Arulselvan A, Commander CW, Pardalos PM, Shylo O (2007) Managing network risk via critical node identification, Risk Management in Telecommunication Networks. Springer, Heidelberg
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 79–91
Bodlaender HL, Hendriks A, Grigoriev A, Grigorieva NV (2010) The valve location problem in simple network topologies. INFORMS Journal on Computing 22(3):433–442
Borgatti SP (2006) Identifying sets of key players in a social network. Computational & Mathematical Organization Theory 12(1):21–34
Chung F, Lu L (2002) Connected components in random graphs with given expected degree sequences. Annals of Combinatorics 6(2):125–145
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):395–411
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):1–25
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):649–680
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):609–619
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.
Erdős P, Rényi A (1959) On random graphs. Publicationes Mathematicae Debrecen 6:290–297
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. 251–262
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 York
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,
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):153–172
Köppe Matthias, Louveaux Quentin, Weismantel Robert (2008) Intermediate integer programming representations using value disjunctions. Discrete Optimization 5(2):293–313
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):396–405
Matisziw TC, Murray AT (2009) Modeling \(s-t\) path availability to support disaster vulnerability assessment of network infrastructure. Computers & Operations Research 36(1):16–26
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):579–589
Newman MEJ (2003) The structure and function of complex networks. SIAM Review 45:167–256
Oosten M, Rutten JHGC, Spieksma FCR (2007) Disconnecting graphs by removing vertices: a polyhedral approach. Statistica Neerlandica 61(1):35–60
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.
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 74:47–97
Resende MGC, Pardalos PM (eds) (2006) Handbook of optimization in telecommunications. Springer,
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):103–119
Shen S, Smith JC, Goli R (2012a) Exact interdiction models and algorithms for disconnecting networks via node deletions. Discrete Optimization 9(3):172–188
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,
Ventresca M (2012) Global search algorithms using a combinatorial unranking-based problem representation for the critical node detection problem. Computers & Operations Research 39(11):2763–2775
Ventresca M, Aleman D (2014) A derandomized approximation algorithm for the critical node detection problem. Computers & Operations Research 43:261–270
Veremyev A., Boginski V., Pasiliao E.L. (2013) Exact identification of critical nodes in sparse networks via new compact formulations. Optimization Letters, pages 1–15
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 9–26
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.
Zachary WW (1977) An information flow model for conflict and fission in small groups. Journal of Anthropological Research 33:452–473
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 184–194
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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Veremyev, A., Prokopyev, O.A. & Pasiliao, E.L. An integer programming framework for critical elements detection in graphs. J Comb Optim 28, 233–273 (2014). https://doi.org/10.1007/s10878-014-9730-4
- Critical node detection
- Critical edge detection
- Network interdiction
- Mixed integer programming