Abstract
The problem of detecting critical elements in a network involves the identification of a subset of elements (nodes, arcs, paths, cliques, etc.) whose deletion minimizes a connectivity measure over the induced network. This problem has attracted significant attention in recent years because of its applications in several fields such as telecommunications, social network analysis, and epidemic control. In this paper we examine the problem of detecting critical cliques (CCP). We first introduce a mathematical formulation for the CCP as an integer linear program. Additionally, we propose a two-stage decomposition strategy that first identifies a candidate clique partition and then uses this partition to reformulate and solve the problem as a generalized critical node problem (GCNP). To generate candidate clique partitions we test two heuristic approaches and solve the resulting (GCNP) using a commercial optimizer. We test our approach in a testbed of 13 instances ranging from 25 to 100 nodes.
This research is partially supported by NSF, DTRA and DURIP grants.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Albert, R., Jeong, H., Barabasi, A.-L.: Error and attack tolerance of complex networks. Nature 406(6794), 378–382 (2000)
Arulselvan, A., Commander, C.W., Elefteriadou, L., Pardalos, P.M.: Detecting critical nodes in sparse graphs. Computers and Operations Research 36(7), 2193–2200 (2009)
Boppana, R., Halldorsson, M.: Approximating maximum independent sets by excluding subgraphs. BIT 32, 180–196 (1992)
Borgatti, S.P.: Identifying sets of key players in a social network. Comput. Math. Organ. Theory 12, 21–34 (2006)
Corley, H., Sha, D.Y.: Most vital links and nodes in weighted networks. Operations Research Letters 1(4), 157–160 (1982)
Dessmark, A., Jansson, J., Lingas, A., Lundell, E.-M., Persson, M.: On the approximability of maximum and minimum edge clique partition problems. International Journal of Foundations of Computer Science 18 (2006, 2007)
Di Summa, M., Grosso, A., Locatelli, M.: Complexity of the critical node problem over trees. Computers and Operations Research 38(12), 1766–1774 (2011)
Dinh, T.N., Xuan, Y., Thai, M.T., Pardalos, P.M., Znati, T.: On new approaches of assessing network vulnerability: Hardness and approximation. IEEE/ACM Transactions on Networking PP(99) (2011)
Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1990)
Grötschel, M., Monma, C., Stoer, M.: Design of survivable networks. In: Ball, C.M.M.O., Magnanti, T.L., Nemhauser, G. (eds.) Network Models. Handbooks in Operations Research and Management Science, vol. 7, pp. 617–672. Elsevier (1995)
Grötschel, M., Wakabayashi, Y.: Facets of the clique partitioning polytope. Mathematical Programming 47, 367–387 (1990)
Grubesic, T.H., Matisziw, T.C., Murray, A.T., Snediker, D.: Comparative approaches for assessing network vulnerability. International Regional Science Review 31(1), 88–112 (2008)
Hopcroft, J., Tarjan, R.: Algorithm 447: efficient algorithms for graph manipulation. Commun. ACM 16, 372–378 (1973)
Houck, D.J., Kim, E., O’Reilly, G.P., Picklesimer, D.D., Uzunalioglu, H.: A network survivability model for critical national infrastructures. Bell Labs Technical Journal 8(4), 153–172 (2004)
Jenelius, E., Petersen, T., Mattsson, L.-G.: Importance and exposure in road network vulnerability analysis. Transportation Research Part A: Policy and Practice 40(7), 537–560 (2006)
Matisziw, T.C., Murray, A.T.: Modeling s-t path availability to support disaster vulnerability assessment of network infrastructure. Comput. Oper. Res. 36, 16–26 (2009)
Oosten, M., Rutten, J.H.G.C., Spieksma, F.C.R.: Disconnecting graphs by removing vertices: a polyhedral approach. Statistica Neerlandica 61(1), 35–60 (2007)
Palmer, C., Steffan, J.: Generating network topologies that obey power laws. In: Global Telecommunications Conference, GLOBECOM 2000, vol. 1, pp. 434–438. IEEE (2000)
Salmeron, J., Wood, K., Baldick, R.: Analysis of electric grid security under terrorist threat. IEEE Transactions on Power Systems 19(2), 905–912 (2004)
Shen, S., Smith, J.C.: Polynomial-time algorithms for solving a class of critical node problems on trees and series-parallel graphs. Networks (2011)
Tao, Z., Zhongqian, F., Binghong, W.: Epidemic dynamics on complex networks. Progress in Natural Science 16(5) (2005)
Wollmer, R.: Removing arcs from a network. Operations Research 12(6), 934–940 (1964)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Walteros, J.L., Pardalos, P.M. (2012). A Decomposition Approach for Solving Critical Clique Detection Problems. In: Klasing, R. (eds) Experimental Algorithms. SEA 2012. Lecture Notes in Computer Science, vol 7276. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30850-5_34
Download citation
DOI: https://doi.org/10.1007/978-3-642-30850-5_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30849-9
Online ISBN: 978-3-642-30850-5
eBook Packages: Computer ScienceComputer Science (R0)