Abstract
This paper develops and compares several heuristic approaches, as well as an exact combinatorial branch-and-bound algorithm, for detecting maximum robust cliques in graphs subjected to multiple uncertain edge failures. The desired robustness properties are enforced using conditional value-at-risk measure. The proposed heuristics are adaptations of the well-known tabu search and GRASP methods, whereas the exact approach is an extension of Östergård’s algorithm for the maximum clique problem. The results of computational experiments on DIMACS graph instances are reported.
Similar content being viewed by others
References
Abello, J., Pardalos, P. M., & Resende, M. G. C. (1999). On maximum clique problems in very large graphs. DIMACS Series, 50, 119–130.
Balas, E., & Xue, J. (1996). Weighted and unweighted maximum clique algorithms with upper bounds from fractional coloring. Algorithmica, 15, 397–412.
Bomze, I. M., Budinich, M., Pardalos, P. M., & Pelillo, M. (1999). The maximum clique problem. In Handbook of Combinatorial Optimization, (pp. 1–74). Springer, New York.
Butenko, S., & Wilhelm, W. E. (2006). Clique-detection models in computational biochemistry and genomics. European Journal of Operational Research, 173(1), 1–17.
Carraghan, R., & Pardalos, P. (1990). An exact algorithm for the maximum clique problem. Operations Research Letters, 9, 375–382.
Corno, F., Prinetto, P., & Sonza Reorda, M. (1995). Using symbolic techniques to find the maximum clique in very large sparse graphs. In Proceedings of the 1995 European conference on Design and Test, EDTC ’95, (pp. 320–324), Washington, DC, USA, 1995. IEEE Computer Society.
Deane, C. M., Salwiński, L., Xenarios, I., & Eisenberg, D. (2002). Protein interactions: two methods for assessment of the reliability of high throughput observations. Molecular & Cellular Proteomics, 1(5), 349–356.
Deng, M., Sun, F., & Chen, T. (2002). Assessment of the reliability of protein–protein interactions and protein function prediction. In Pacific Symposium on Biocomputing (PSB 2003), (pp. 140–151), 2002.
DIMACS. NP Hard Problems: Maximum Clique, Graph Coloring, and Satisfiability. The Second DIMACS Implementation Challenge. http://dimacs.rutgers.edu/Challenges/, 1992–1993.
DIMACS. Algorithm Implementation Challenge: Graph Partitioning and Graph Clustering. The Tenth DIMACS Implementation Challenge. http://dimacs.rutgers.edu/Challenges/, 2012.
Feo, T. A., & Resende, M. G. C. (1989). A probabilistic heuristic for a computationally difficult set covering problem. Operations Research Letters, 8, 67–71.
Feo, T. A., & Resende, M. G. C. (1995). Greedy randomized adaptive search procedures. Journal of Global Optimization, 6, 109–133.
Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486(3), 75–174.
Friden, C., Hertz, A., & de Werra, D. (1989). STABULUS: A technique for finding stable sets in large graphs with tabu search. Computing, 42, 35–44.
Gendreau, M., Soriano, P., & Salvail, L. (1993). Solving the maximum clique problem using a tabu search approach. Annals of Operations Research, 41, 385–403.
Glover, F. (1986). Future paths for integer programming and links to artificial intelligence. Computers and Operations Research, 13(5), 533–549.
Glover, F. (1989). Tabu search. Part I. ORSA Journal on Computing, 1(3), 190–206.
Glover, F. (1990). Tabu search. Part II. ORSA Journal on Computing, 2(1), 4–32.
Goldberg, D. S., & Roth, F. P. (2003). Assessing experimentally derived interactions in a small world. Proceedings of the National Academy of Sciences, 100(8), 4372–4376.
Hintsanen, P. (2007). The most reliable subgraph problem. In Knowledge Discovery in Databases: PKDD 2007, (pp. 471–478). Springer, 2007.
Hintsanen, P., & Toivonen, H. (2008). Finding reliable subgraphs from large probabilistic graphs. Data Mining and Knowledge Discovery, 17(1), 3–23.
Jin, R., Liu, L., & Aggarwal, C. (2011). Discovering highly reliable subgraphs in uncertain graphs. In Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, (pp. 992–1000). ACM, 2011.
Kollios, G., Potamias, M., & Terzi, E. (2013). Clustering large probabilistic graphs. IEEE Transactions on Knowledge and Data Engineering, 25(2), 325–336.
Krokhmal, P., Murphey, R., Pardalos, P. M., Uryasev, S., & Zrazhevski, G. (2003). Robust decision making: Addressing uncertainties in distributions. In S. Butenko, R. Murphey, & P. M. Pardalos (Eds.), Cooperative Control: Models, Applications and Algorithms, volume 1 of Cooperative Systems (pp. 165–185). New York: Springer.
Li, X., Wu, M., Kwoh, C.-K., & Ng, S.-K. (2010). Computational approaches for detecting protein complexes from protein interaction networks: a survey. BMC Genomics, 11(Suppl 1), S3.
Liu, L., Jin, R., Aggarwal, C., & Shen, Y. (2012). Reliable clustering on uncertain graphs. In ICDM, (pp. 459–468). Citeseer, 2012.
Luce, R., & Perry, A. (1949). A method of matrix analysis of group structure. Psychometrika, 14, 95–116.
Miao, Z., Balasundaram, B., & Pasiliao, E. L. (2014). An exact algorithm for the maximum probabilistic clique problem. Journal of Combinatorial Optimization, 28, 105–120.
Östergård, P. R. J. (2002). A fast algorithm for the maximum clique problem. Discrete Applied Mathematics, 120, 197–207.
Palmquist, J., Krohmal, P., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. The Journal of Risk, 2, 11–27.
Pattillo, J., Youssef, N., & Butenko, S. (2013). On clique relaxation models in network analysis. European Journal of Operational Research, 226, 9–18.
Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk, 2, 21–42.
Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26(7), 1443–1471.
Rysz, M., Mirghorbani, M., Krokhmal, P., & Pasiliao, E. L. (2014). On risk-averse maximum weighted subgraph problems. Journal of Combinatorial Optimization, 28(1), 167–185.
Seidman, S. B., & Foster, B. L. (1978). A graph theoretic generalization of the clique concept. Journal of Mathematical Sociology, 6, 139–154.
Soriano, P., & Gendreau, M. (1996). Diversification strategies in tabu search algorithms for the maximum clique problem. Annals of Operations Research, 63, 189–207.
Spirin, V., & Mirny, L. A. (2003). Protein complexes and functional modules in molecular networks. Proceedings of the National Academy of Sciences, 100(21), 12123–12128.
Tomita, E., & Seki, T. (2003). An efficient branch-and-bound algorithm for finding a maximum clique. In C. Calude, M. Dinneen, & V. Vajnovszki (Eds.), Discrete mathematics and theoretical computer science, volume 2731 of lecture notes in computer science (pp. 278–289). Berlin: Springer.
Verma, A., Buchanan, A., & Butenko, S. (2015). Solving the maximum clique and vertex coloring problems on very large sparse networks. INFORMS Journal on Computing, 27, 164–177.
Wu, Q., & Hao, J. K. (2013). An adaptive multistart tabu search approach to solve the maximum clique problem. Journal of Combinatorial Optimization, 26, 86–108.
Wu, Q., & Hao, J. K. (2015). A review on algorithms for maximum clique problems. European Journal of Operational Research, 242(3), 693–709.
Wu, Q., Hao, J. K., & Glover, F. (2012). Multi-neighborhood tabu search for the maximum weight clique problem. Annals of Operations Research, 196, 611–634.
Yannakakis, M. (1978). Node-and edge-deletion np-complete problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, 1978.
Yu, H., Paccanaro, A., Trifonov, V., & Gerstein, M. (2006). Predicting interactions in protein networks by completing defective cliques. Bioinformatics, 22, 823–829.
Zou, Z., Li, J., Gao, H., & Zhang, S. (2010). Finding top-\(k\) maximal cliques in an uncertain graph. In Proceedings of the 26th IEEE International Conference on Data Engineering (ICDE), (pp. 649–652). IEEE, 2010.
Acknowledgments
The authors would like to thank two anonymous referees whose suggestions helped to improve the paper. This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. Partial support by AFOSR under Grants FA9550-12-1-0103 and FA8651-12-2-0011, as well as the U.S. Department of Energy grant DE-SC0002051 is also gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yezerska, O., Butenko, S. & Boginski, V.L. Detecting robust cliques in graphs subject to uncertain edge failures. Ann Oper Res 262, 109–132 (2018). https://doi.org/10.1007/s10479-016-2161-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-016-2161-0