A GPU based local search algorithm for the unweighted and weighted maximum s-plex problems

  • Bruno NogueiraEmail author
  • Rian G. S. Pinheiro
Original Research


Given a graph G(VE) and a value \(s \in {\mathbb {N}}\), an s-plex S is a subset of V such that each vertex \(v \in S\) has at least \(|S|-s\) adjacent vertices in the subgraph induced by S. This work proposes a GPU based local search heuristic, called GPULS, for the problems of finding an s-plex of maximum cardinality and finding an s-plex of maximum weight. The proposed heuristic works well on both problems without any modification on its parameters or its code. GPULS considers two neighborhood structures, which are explored using tabu search and a first-improvement approach. We compare GPULS with the current best-performing exact methods and heuristics. The results obtained by GPULS are highly competitive, even when it runs on a CPU-only architecture. Moreover, we observed speedups of up to 16 times by running the heuristic on a hybrid CPU–GPU architecture.


Clique relaxation Maximum s-plex Maximum clique GPU Tabu search Metaheuristics 


Supplementary material

10479_2019_3159_MOESM1_ESM.pdf (129 kb)
Supplementary material 1 (pdf 129 KB)


  1. Balasundaram, B. (2007). Graph theoretic generalizations of clique: Optimization and extensions. Ph.D. thesis.Google Scholar
  2. Balasundaram, B., Butenko, S., & Hicks, I. V. (2011). Clique relaxations in social network analysis: The maximum k-plex problem. Operations Research, 59(1), 133–142.CrossRefGoogle Scholar
  3. Boginski, V., Butenko, S., Shirokikh, O., Trukhanov, S., & Lafuente, J. G. (2014). A network-based data mining approach to portfolio selection via weighted clique relaxations. Annals of Operations Research, 216(1), 23–34.CrossRefGoogle Scholar
  4. Carraghan, R., & Pardalos, P. M. (1990). An exact algorithm for the maximum clique problem. Operations Research Letters, 9(6), 375–382.CrossRefGoogle Scholar
  5. Corstjens, J., Dang, N., Depaire, B., Caris, A., & De Causmaecker, P. (2018). A combined approach for analysing heuristic algorithms. Journal of Heuristics,. Scholar
  6. da Silva, M. R. C., Tavares, W. A., Dias, F. C. S., & Neto, M. B. C. (2017). Algoritmo branch-and-bound para o problema do k-plex máximo. In: Anais do XLIX Simpsio Brasileiro de Pesquisa OperacionalGoogle Scholar
  7. Gendreau, M., & Potvin, J. Y. (2010). Handbook of metaheuristics (pp. 41–60). Berlin: Springer.Google Scholar
  8. Gschwind, T., Irnich, S., & Podlinski, I. (2016). Maximum weight relaxed cliques and Russian doll search revisited. Discrete Applied Mathematics, 234, 131–138. Scholar
  9. Gujjula, K. R., Seshadrinathan, K. A., & Meisami, A. (2014). A hybrid metaheuristic for the maximum k-plex problem. In NATO advanced research workshop on examining robustness and vulnerability of critical infrastructure networks, IOS Google Scholar
  10. Harris, M. (2007). Optimizing Parallel Reduction in CUDA. NVIDIA Developer Technology.
  11. Komusiewicz, C. (2016). Multivariate algorithmics for finding cohesive subnetworks. Algorithms, 9(1), 21.CrossRefGoogle Scholar
  12. Martí, R., Moreno-Vega, J. M., & Duarte, A. (2010). Advanced multi-start methods. In M. Gendreau & J. Y. Potvin (Eds.), Handbook of metaheuristics (pp. 265–281). Boston, MA: Springer. Scholar
  13. McClosky, B. (2008). Independence systems and stable set relaxations. Ph.D. thesis, Rice UniversityGoogle Scholar
  14. McClosky, B., & Hicks, I. V. (2012). Combinatorial algorithms for the maximum k-plex problem. Journal of combinatorial optimization, 23(1), 29–49.CrossRefGoogle Scholar
  15. Miao, Z., & Balasundaram, B. (2017). Approaches for finding cohesive subgroups in large-scale social networks via maximum k-plex detection. Networks, 69(4), 388–407.CrossRefGoogle Scholar
  16. Nogueira, B., & Pinheiro, R. G. S. (2018). A cpu–gpu local search heuristic for the maximum weight clique problem on massive graphs. Computers and Operations Research, 90, 232–248. Scholar
  17. Nogueira, B., Pinheiro, R. G. S., & Subramanian, A. (2018). A hybrid iterated local search heuristic for the maximum weight independent set problem. Optimization Letters, 12(3), 567–583. Scholar
  18. Östergård, P. R. (2002). A fast algorithm for the maximum clique problem. Discrete Applied Mathematics, 120(1), 197–207.CrossRefGoogle Scholar
  19. Pattillo, J., Youssef, N., & Butenko, S. (2013). On clique relaxation models in network analysis. European Journal of Operational Research, 226(1), 9–18.CrossRefGoogle Scholar
  20. Pullan, W., & Hoos, H. H. (2006). Dynamic local search for the maximum clique problem. Journal of Artificial Intelligence Research, 25, 159–185.CrossRefGoogle Scholar
  21. Seidman, S. B., & Foster, B. L. (1978). A graph-theoretic generalization of the clique concept. Journal of Mathematical sociology, 6(1), 139–154.CrossRefGoogle Scholar
  22. 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.CrossRefGoogle Scholar
  23. Wang, Y., Cai, S., & Yin, M. (2016). Two efficient local search algorithms for maximum weight clique problem. In AAAI conference on artificial intelligence,
  24. Wu, Q., & Hao, J. K. (2013). An adaptive multistart tabu search approach to solve the maximum clique problem. Journal of Combinatorial Optimization, 26(1), 86–108.CrossRefGoogle Scholar
  25. Wu, Q., Hao, J. K., & Glover, F. (2012). Multi-neighborhood tabu search for the maximum weight clique problem. Annals of Operations Research, 196(1), 611–634.CrossRefGoogle Scholar
  26. Xiao, M., Lin, W., Dai, Y., & Zeng, Y. (2017). A fast algorithm to compute maximum k-plexes in social network analysis. In AAAI conference on artificial intelligence (pp 919–925)Google Scholar
  27. Xiao, S., & Feng, W. (2010). Inter-block gpu communication via fast barrier synchronization. In: 2010 IEEE international symposium on parallel and distributed processing (IPDPS) (pp 1–12). IEEE.Google Scholar
  28. Zhou, Y., & Hao, J. K. (2017). Frequency-driven tabu search for the maximum s-plex problem. Computers and Operations Research, 86, 65–78.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Universidade Federal de AlagoasMaceióBrazil

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