Abstract
We consider the Critical Node Problem: given an undirected graph and an integer number K, at most K nodes have to be deleted from the graph in order to minimize a connectivity measure in the residual graph. We combine the basic steps used in common greedy algorithms with some flavour of local search, in order to obtain simple hybrid heuristic algorithms. The obtained algorithms are shown to be effective, delivering improved performances (solution quality and speed) with respect to known greedy algorithms and other more sophisticated state of the art methods.
This is a preview of subscription content, log in via an institution to check access.
Notes
Calculated as
$$\begin{aligned} \frac{A(I)-\text {BEST}(I)}{\text {BEST}(I)} \end{aligned}$$
References
Addis, B., Di Summa, M., & Grosso, A. (2013). Removing critical nodes from a graph: Complexity results and polynomial algorithms for the case of bounded treewidth. Discrete Applied Mathematics, 16–17, 2349–2360.
Aringhieri, R., Grosso, A., Hosteins, P., & Scatamacchia, R. (2015). VNS solutions for the critical node problem. In Proceedings of the VNS’14 conference. Electronic notes in discrete mathematics (Vol. 47, pp. 37–44).
Arulselvan, A., Commander, C. W., Elefteriadou, L., & Pardalos, P. M. (2009). Detecting critical nodes in sparse graphs. Computers & Operations Research, 36, 2193–2200.
Boginski, V., & Commander, C. W. (2009). Identifying critical nodes in protein–protein interaction networks. In S. Butenko, W. A. Chaovalitwongse, & P. M. Pardalos (Eds.), Clustering challenges in biological networks (pp. 153–168). Singapore: World Scientific Publishing.
Borgatti, S. P. (2006). Identifying sets of key players in a network. Computational and Mathematical Organization Theory, 12, 21–34.
Brandes, U. (2001). A faster algorithm for betweenness centrality. Journal of Mathematical Sociology, 25, 163–177.
Di Summa, M., Grosso, A., & Locatelli, M. (2011). The critical node problem over trees. Computers and Operations Research, 38, 1766–1774.
Di Summa, M., Grosso, A., & Locatelli, M. (2012). Branch and cut algorithms for detecting critical nodes in undirected graphs. Computational Optimization and Applications, 53, 649–680.
Dinh, T., Xuan, Y., Thai, M., Pardalos, P., & Znati, T. (2012). On new approaches of assessing network vulnerability: Hardness and approximation. IEEE/ACM Transactions on Networking, 20, 609–619.
Dinh, T. N., & Thai, M. T. (2011). Precise structural vulnerability assessment via mathematical programming. In MILCOM 2011–2011 IEEE military communications conference (pp. 1351–1356).
Dolan, E., & Moré, J. (2002). Benchmarking optimization software with performance profiles. Mathematical Programming, 91(2), 201–13.
Edalatmanesh, M. (2013). Heuristics for the critical node detection problem in large complex networks. Ph.D. thesis, Faculty of Mathematics and Science, Brock University, St. Catharines, ON.
Golden, B. L., & Shier, D. R. (Eds.) (2014). Network interdiction applications and extensions. Virtual Issue on Networks. http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0037/homepage/virtual_issue_-_network_interdiction_applications_and_extensions.htm.
Hopcroft, J., & Tarjan, R. (1973). Algorithm 447: Efficient algorithms for graph manipulation. Communications of the ACM, 16(6), 372–378.
Papadimitriou, C., & Steiglitz, K. (1982). Combinatorial optimization: Algorithms and complexity. Englewood Cliffs, NJ: Prentice-Hall.
Shen, S., & Smith, J. (2012). Polynomial-time algorithms for solving a class of critical node problems on trees and series–parallel graphs. Networks, 60(2), 103–119. doi:10.1002/net.20464.
Shen, S., Smith, J., & Goli, R. (2012). Exact interdiction models and algorithms for disconnecting networks via node deletions. Discrete Optimization, 9, 172–88.
Ventresca, M. (2012). Global search algorithms using a combinatorial unranking-based problem representation for the critical node detection problem. Computers & Operations Research, 39, 2763–2775.
Ventresca, M., & Aleman, D. (2014). A derandomized approximation algorithm for the critical node detection problem. Computers and Operations Research, 43, 261–270.
Ventresca, M., & Aleman, D. (2015). Efficiently identifying critical nodes in large complex networks. Computational Social Networks, 2(1), 6. doi:10.1186/s40649-015-0010-y.
Veremyev, A., Boginski, V., & Pasiliao, E. (2014). Exact identification of critical nodes in sparse networks via new compact formulations. Optimization Letters, 8, 1245–1259.
Veremyev, A., Prokopyev, O., & Pasiliao, E. (2014). An integer programming framework for critical elements detection in graphs. Journal of Combinatorial Optimization, 28, 233–273.
Veremyev, A., Prokopyev, O., & Pasiliao, E. (2015). Critical nodes for distance-based connectivity and related problems in graphs. Networks, 66, 170–195.
Walteros, J., & Pardalos, P. (2012). Selected topics in critical element detection. In N. J. Daras (Ed.), Applications of mathematics and informatics in military science, Springer optimization and its applications (Vol. 71, pp. 9–26). New York: Springer. doi:10.1007/978-1-4614-4109-0_2.
Wollmer, R. (1964). Removing arcs from a network. Operations Research, 12, 934–940.
Wood, R. K. (1993). Deterministic network interdiction. Mathematical and Computer Modelling, 17, 1–18.
Author information
Authors and Affiliations
Corresponding author
Additional information
Work supported by a Google Focused Grant on Mathematical Programming, project “Exact and Heuristic Algorithms for Detecting Critical Nodes in Graphs”.
Rights and permissions
About this article
Cite this article
Addis, B., Aringhieri, R., Grosso, A. et al. Hybrid constructive heuristics for the critical node problem. Ann Oper Res 238, 637–649 (2016). https://doi.org/10.1007/s10479-016-2110-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-016-2110-y