The triangle k-club problem

Abstract

Graph models have long been used in social network analysis and other social and natural sciences to render the analysis of complex systems easier. In applied studies, to understand the behaviour of social networks and the interactions that command that behaviour, it is often necessary to identify sets of elements which form cohesive groups, i.e., groups of actors that are strongly interrelated. The clique concept is a suitable representation for groups of actors that are all directly related pair-wise. However, many social relationships are established not only face-to-face but also through intermediaries, and the clique concept misses all the latter. To deal with these cases, it is necessary to adopt approaches that relax the clique concept. In this paper we introduce a new clique relaxation—the triangle k-club—and its associated maximization problem—the maximum triangle k-club problem. We propose integer programming formulations for the problem, stated in different variable spaces, and derive valid inequalities to strengthen their linear programming relaxations. Computational results on randomly generated and real-world graphs, with \(k=2\) and \(k=3\), are reported.

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References

  1. Alba RD (1973) A graph-theoretic definition of a sociometric clique. J Mater Sociol 3:113–126

    MathSciNet  Article  MATH  Google Scholar 

  2. Alderson DL (2008) Catching the “network science” bug: insight and opportunity for the operations researcher. Oper Res 56:1047–1065

    MathSciNet  Article  MATH  Google Scholar 

  3. Almeida MT, Carvalho FD (2012) Integer models and upper bounds for the 3-club problem. Networks 60:155–166

    MathSciNet  Article  MATH  Google Scholar 

  4. Almeida MT, Carvalho FD (2014) An analytical comparison of the LP relaxations of integer models for the \(k\)-club problem. Eur J Oper Res 232:489–498

    MathSciNet  Article  MATH  Google Scholar 

  5. Balas E (2005) Projection, lifting and extended formulation in integer and combinatorial optimization. Ann Oper Res 140:125–161

    MathSciNet  Article  MATH  Google Scholar 

  6. Balasundaram B, Butenko S, Trukhanov S (2005) Novel approaches for analyzing biological networks. J Comb Optim 10:23–39

    MathSciNet  Article  MATH  Google Scholar 

  7. Balasundaram B, Butenko S, Hicks IV (2011) Clique relaxations in social network analysis: the maximum \(k\)-plex problem. Oper Res 59:133–142

    MathSciNet  Article  MATH  Google Scholar 

  8. Batagelj V, Mrvar A (1998) Pajek: a program for large network analysis. Connections 21(2):47–57

    Google Scholar 

  9. Blażewicz J, Formanowicz P, Kasprak M (2005) Selected combinatorial problems of computational biology. Eur J Oper Res 161:585–597

    MathSciNet  Article  MATH  Google Scholar 

  10. Boginski V, Butenko S, Pardalos PM (2006) Mining market data: a network approach. Comput Oper Res 33:3171–3184

    Article  MATH  Google Scholar 

  11. Bourjolly J-M, Laporte G, Pesant G (2000) Heuristics for finding \(k\)-clubs in an undirected graph. Comput Oper Res 27:559–569

    MathSciNet  Article  MATH  Google Scholar 

  12. Bourjolly J-M, Laporte G, Pesant G (2002) An exact algorithm for the maximum \(k-\)club problem in an undirected graph. Eur J Oper Res 138:21–28

    MathSciNet  Article  MATH  Google Scholar 

  13. Carvalho FD, Almeida MT (2011) Upper bounds and heuristics for the 2-club problem. Eur J Oper Res 210:489–494

    MathSciNet  Article  MATH  Google Scholar 

  14. Cavique L (2007) A scalable algorithm for the market basket analysis. J Retail Consum Serv 14:400–407

    Article  Google Scholar 

  15. Diestel R (2006) Graph theory. Graduate texts in mathematics, 173. Springer, New York

    Google Scholar 

  16. Luce RD (1950) Connectivity and generalized cliques in sociometric group structure. Psychometrika 15:169–190

    MathSciNet  Article  Google Scholar 

  17. Mahdavi Pajouh F, Balasundaram B (2012) On inclusionwise maximal and maximum cardinality \(k\)-clubs in graphs. Discret Optim 9:84–97

    MathSciNet  Article  MATH  Google Scholar 

  18. Mokken RJ (1979) Cliques, clubs and clans. Qual Quant 13:161–173

    Article  Google Scholar 

  19. Moradi E, Balasundaram B (2015) Finding a maximum \(k\)-club using the \(k\)-clique formulation and canonical hypercube cuts. Optim Lett. doi:10.1007/s11590-015-0971-7

    Google Scholar 

  20. Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley, New York

    Google Scholar 

  21. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167–256

    MathSciNet  Article  MATH  Google Scholar 

  22. Pattillo J, Veremyev A, Butenko S, Boginski V (2013a) On the maximum quasi-clique problem. Discret Appl Math 161:244–257

    MathSciNet  Article  MATH  Google Scholar 

  23. Pattillo J, Youssef N, Butenko S (2013b) On clique relaxation models in network analysis. Eur J Oper Res 226:9–18

    MathSciNet  Article  MATH  Google Scholar 

  24. Veremyev A, Boginski V (2012) Identifying large robust network clusters via new compact formulations of maximum \(k\)-club problems. Eur J Oper Res 218:316–326

    MathSciNet  Article  MATH  Google Scholar 

  25. Veremyev A, Prokopyev O, Pasiliao E (2015) Critical nodes for distance-based connectivity and related problems in graphs. Networks 66:170–195

    MathSciNet  Article  Google Scholar 

  26. Wasserman S, Faust K (1994) Social network analysis. Cambridge University Press, New York

    Google Scholar 

  27. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442

    Article  Google Scholar 

  28. Wotzlaw A (2014) On solving the maximum \(k\)-club problem. arXiv:1403.5111v2

  29. Wu Q, Hao J-K (2015) A review on algorithms for maximum clique problems. Eur J Oper Res 242:693–709

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

This work is supported by Fundação para a Ciência e a Tecnologia, under the Project: UID/MAT/04561/2013. We thank the three anonymous referees for their valuable comments and suggestions which helped to improve the paper and opened interesting paths for future work. We also thank Ann Henshall, who was always ready to elucidate our English grammar and vocabulary doubts.

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Correspondence to Filipa D. Carvalho.

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Carvalho, F.D., Almeida, M.T. The triangle k-club problem. J Comb Optim 33, 814–846 (2017). https://doi.org/10.1007/s10878-016-0009-9

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Keywords

  • Clique relaxations
  • Integer formulations
  • Valid inequalities
  • Cliques
  • Social network analysis