Skip to main content
SpringerLink
Log in

Identifying risk-averse low-diameter clusters in graphs with stochastic vertex weights

  • S.I.: Risk Management Approaches in Engineering Applications
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

In this work, we study the problem of detecting risk-averse low-diameter clusters in graphs. It is assumed that the clusters represent k-clubs and that uncertain information manifests itself in the form of stochastic vertex weights whose joint distribution is known. The goal is to find a k-club of minimum risk contained in the graph. A stochastic programming framework that is based on the formalism of coherent risk measures is used to quantify the risk of a cluster. We show that the selected representation of risk guarantees that the optimal subgraphs are maximal clusters. A combinatorial branch-and-bound algorithm is proposed and its computational performance is compared with an equivalent mathematical programming approach for instances with \(k=2,3,\) and 4.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. “Expectation-boundedness” is also known as “aversity” [36], but we use the former term in this work so as to avoid semantic confusion when referring to “risk-averse” subgraphs.

  2. Indicated in Algorithm 1 by the assignment “\(\text{ fathom } := \text{ True }\)”.

References

  1. Abello, J., Pardalos, P., & Resende, M. (1999). On maximum clique problems in very large graphs. In Abello, J. and Vitter, J. (Eds.) External memory algorithms and visualization, volume 50 of DIMACS Series on Discrete Mathematics and Theoretical Computer Science, pp. 119–130. Providence, RI: American Mathematical Society.

  2. Abello, J., Resende, M., & Sudarsky, S. (2002). Massive quasi-clique detection. In S. Rajsbaum (Ed.), LATIN 2002: Theoretical informatics (pp. 598–612). London: Springer.

    Chapter  Google Scholar 

  3. Alba, R. D. (1973). A graph-theoretic definition of a sociometric clique. Journal of Mathematical Sociology, 3, 3–113.

    Article  Google Scholar 

  4. Aneja, Y. P., Chandrasekaran, R., & Nair, K. P. K. (2001). Maximizing residual flow under an arc destruction. Networks, 38(4), 194–198.

    Article  Google Scholar 

  5. Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.

    Article  Google Scholar 

  6. Atamtrk, A., & Zhang, M. (2007). Two-stage robust network flow and design under demand uncertainty. Operations Research, 55(4), 662–673.

    Article  Google Scholar 

  7. Babel, L. (1994). A fast algorithm for the maximum weight clique problem. Computing, 52(1), 31–38.

    Article  Google Scholar 

  8. Balas, E., & Yu, C. S. (1986). Finding a maximum clique in an arbitrary graph. SIAM Journal on Computing, 15(4), 1054–1068.

    Article  Google Scholar 

  9. Balasundaram, B., Butenko, S., & Trukhanov, S. (2005). Novel approaches for analyzing biological networks. Journal of Combinatorial Optimization, 10(1), 23–39.

    Article  Google Scholar 

  10. Balasundaram, B., & Pajouh, F. M. (2013). Graph theoretic clique relaxations and applications. In P. M. Pardalos, D.-Z. Du, & R. Graham (Eds.), Handbook of combinatorial optimization (2nd ed., pp. 1559–1598). Berlin: Springer.

    Chapter  Google Scholar 

  11. Barabasi, A. (2012). Network science. Center for Complex Network Research at Northeastern University (http://barabasilab.neu.edu/networksciencebook/downlPDF.html), Boston, MA.

  12. Boginski, V., Butenko, S., & Pardalos, P. (2006). Mining market data: A network approach. Computers & Operations Research, 33(11), 3171–3184.

    Article  Google Scholar 

  13. Bourjolly, J.-M., Laporte, G., & Pesant, G. (2002). An exact algorithm for the maximum k-club problem in an undirected graph. European Journal of Operational Research, 138(1), 21–28.

    Article  Google Scholar 

  14. Campbell, A. M., & Thomas, B. W. (2008). Probabilistic traveling salesman problem with deadlines. Transportation Science, 42(1), 1–21.

    Article  Google Scholar 

  15. Carmo, R., & Zge, A. (2012). Branch and bound algorithms for the maximum clique problem under a unified framework. Journal of the Brazilian Computer Society, 18(2), 137–151.

    Article  Google Scholar 

  16. Carraghan, R., & Pardalos, P. M. (1990). An exact algorithm for the maximum clique problem. Operations Research Letters, 9(6), 375–382.

    Article  Google Scholar 

  17. Chang, M.-S., Hung, L.-J., Lin, C.-R., & Su, P.-C. (2013). Finding large k-clubs in undirected graphs. Computing, 95(9), 739–758.

    Article  Google Scholar 

  18. Cook, D. J., & Holder, L. B. (2000). Graph-based data mining. IEEE Intelligent Systems, 15(2), 32–41.

    Article  Google Scholar 

  19. Delbaen, F. (2002). Coherent risk measures on general probability spaces, pp. 1–37.

  20. Glockner, G. D., & Nemhauser, G. L. (2000). A dynamic network flow problem with uncertain arc capacities: Formulation and problem structure. Operations Research, 48(2), 233–242.

    Article  Google Scholar 

  21. Gupta, A., Nagarajan, V., & Ravi, R. (2012). Technical note approximation algorithms for vrp with stochastic demands. Operations Research, 60(1), 123–127.

    Article  Google Scholar 

  22. Hill, S., Provost, F., & Volinsky, C. (2006). Network-based marketing: Identifying likely adopters via consumer networks. Statistical Science, 22, 256–275.

    Article  Google Scholar 

  23. Iacobucci, D., & Hopkins, N. (1992). Modeling dyadic interactions and networks in marketing. Journal of Marketing Research, 24, 5–17.

    Article  Google Scholar 

  24. Konc, J., & Janezic, D. (2007). An improved branch and bound algorithm for the maximum clique problem. Proteins, 4, 5.

    Google Scholar 

  25. Krokhmal, P., & Soberanis, P. (2010). Risk optimization with \(p\)-order conic constraints: A linear programming approach. European Journal of Operational Research, 301(3), 653–671.

    Article  Google Scholar 

  26. Krokhmal, P., Zabarankin, M., & Uryasev, S. (2011). Modeling and optimization of risk. Surveys in Operations Researh and Management Science, 16(2), 49–66.

    Article  Google Scholar 

  27. Krokhmal, P. A. (2007). Higher moment coherent risk measures. Quantitative Finance, 7, 373–387.

    Article  Google Scholar 

  28. Kumlander, D. (2004). A new exact algorithm for the maximum-weight clique problem based on a heuristic vertex-coloring and a backtrack search. In Proceedings of the Fourth International conference on engineering computational technology, pp. 137–138. Civil-Comp Press.

  29. Luce, R. (1950). Connectivity and generalized cliques in sociometric group structure. Psychometrika, 15(2), 169–190.

    Article  Google Scholar 

  30. Mokken, R. (1979). Cliques, clubs and clans. Quality and Quantity, 13(2), 161–173.

    Article  Google Scholar 

  31. Morenko, Y., Vinel, A., Yu, Z., & Krokhmal, P. (2013). On \(p\)-cone linear discrimination. European Journal of Operational Research, 231(3), 784–789.

    Article  Google Scholar 

  32. Östergård, P. R. J. (2001). A new algorithm for the maximum-weight clique problem. Nordic Journal of Computing, 8(4), 424–436.

    Google Scholar 

  33. Östergård, P. R. J. (2002). A fast algorithm for the maximum clique problem. Discrete Applied Mathematics, 120(1–3), 197–207. (Special Issue devoted to the 6th Twente Workshop on Graphs and Combinatorial Optimization).

    Article  Google Scholar 

  34. Pajouh, F. M., & Balasundaram, B. (2012). On inclusionwise maximal and maximum cardinality k-clubs in graphs. Discrete Optimization, 9(2), 84–97.

    Article  Google Scholar 

  35. Pattillo, J., Veremyev, A., Butenko, S., & Boginski, V. (2013). On the maximum quasi-clique problem. Discrete Applied Mathematics, 161(1–2), 244–257.

    Article  Google Scholar 

  36. Rockafellar, R. T., & Uryasev, S. (2013). The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surveys in Operations Research and Management Science, 18, 33–53.

    Article  Google Scholar 

  37. Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2006). Generalized deviations in risk analysis. Finance and Stochastics, 10(1), 51–74.

    Article  Google Scholar 

  38. Rysz, M., Mirghorbani, M., Krokhmal, P., & Pasiliao, E. (2014). On risk-averse maximum weighted subgraph problems. Journal of Combinatorial Optimization, 28(1), 167–185.

    Article  Google Scholar 

  39. Schfer, A., Komusiewicz, C., Moser, H., & Niedermeier, R. (2012). Parameterized computational complexity of finding small-diameter subgraphs. Optimization Letters, 6(5), 883–891.

    Article  Google Scholar 

  40. Seidman, S. B., & Foster, B. L. (1978). A graph theoretic generalization of the clique concept. Journal of Mathematical Sociology, 6, 139–154.

    Article  Google Scholar 

  41. Tomita, E., Sutani, Y., Higashi, T., Takahashi, S., & Wakatsuki, M. (2010). A simple and faster branch-and-bound algorithm for finding a maximum clique. In M. Rahman & S. Fujita (Eds.), WALCOM: Algorithms and Computation (Vol. 5942, pp. 191–203)., Lecture Notes in Computer Science Berlin Heidelberg: Springer.

    Chapter  Google Scholar 

  42. Trukhanov, S., Balasubramaniam, C., Balasundaram, B., & Butenko, S. (2013). Algorithms for detecting optimal hereditary structures in graphs, with application to clique relaxations. Computational Optimization and Applications, 56(1), 113–130.

    Article  Google Scholar 

  43. Veremyev, A., Prokopyev, O., and Pasiliao, E. (2014). Critical nodes for communication efficiency and related problems in graphs. Working Paper.

  44. Verweij, B., Ahmed, S., Kleywegt, A., Nemhauser, G., & Shapiro, A. (2003). The sample average approximation method applied to stochastic routing problems: A computational study. Computational Optimization and Applications, 24(2–3), 289–333.

    Article  Google Scholar 

  45. Vinel, A., & Krokhmal, P. (2014). Polyhedral approximations in \(p\)-order cone programming. Optimization Methods and Software, 29(6), 1210–1237.

    Article  Google Scholar 

  46. Woodside, A. G., & DeLozier, M. W. (1976). Effects of word of mouth advertising on consumer risk taking. Journal of Advertising, 5(4), 12–19.

    Article  Google Scholar 

  47. Yannakakis, M. (1978). Node-and edge-deletion np-complete problems. In STOC’78: Proceedings of the 10th annual ACM symposium on theory of computing, pp. 253–264, New York: ACM Press.

Download references

Acknowledgments

This research was performed while the first author held a National Research Council Research Associateship Award at the Air Force Research Laboratory. This work was supported in part by the AFOSR grant FA9550-12-1-0142, DTRA grant HDTRA1-14-1-0065, and the U.S. Department of Air Force Grant FA8651-14-2-0003. In addition, support by the AFRL Mathematical Modeling and Optimization Institute is gratefully acknowledged.

Author information

Authors and Affiliations

  1. National Research Council, Air Force Research Laboratory, 101 West Eglin Blvd., Eglin AFB, FL, 32542, USA

    Maciej Rysz

  2. Management Science and Information Systems Department, University of Massachusetts Boston, 100 Morrissey Blvd., Boston, MA, 02125, USA

    Foad Mahdavi Pajouh

  3. Department of Systems and Industrial Engineering, University of Arizona, 1127 E James E. Rogers Way, Tucson, AZ, 85721, USA

    Pavlo Krokhmal

  4. Munitions Directorate, Air Force Research Laboratory, 101 West Eglin Blvd., Eglin AFB, FL, 32542, USA

    Eduardo L. Pasiliao

Authors
  1. Maciej Rysz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Foad Mahdavi Pajouh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pavlo Krokhmal
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eduardo L. Pasiliao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavlo Krokhmal.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rysz, M., Pajouh, F.M., Krokhmal, P. et al. Identifying risk-averse low-diameter clusters in graphs with stochastic vertex weights. Ann Oper Res 262, 89–108 (2018). https://doi.org/10.1007/s10479-016-2212-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-016-2212-6

Keywords

Search

Navigation