Detecting robust cliques in graphs subject to uncertain edge failures

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.

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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.

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Correspondence to Sergiy Butenko.

Appendix

Appendix

See Tables 5 and 6.

Table 5 Values of \({ CVaR}_{0.9}[L_C]\) for various sizes of clique C, where each edge fails with the same probability p
Table 6 Value of \({ CVaR}_{0.95}[L_C]\) for various sizes of clique C, where each edge fails with the same probability p

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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

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Keywords

  • Maximum clique
  • Robust clique
  • Conditional value-at-risk
  • Heuristic
  • Tabu search
  • GRASP