Skip to main content
SpringerLink
Log in

Exact MIP-based approaches for finding maximum quasi-cliques and dense subgraphs

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Given a simple graph and a constant \(\gamma \in (0,1]\), a \(\gamma \)-quasi-clique is defined as a subset of vertices that induces a subgraph with an edge density of at least \(\gamma \). This well-known clique relaxation model arises in a variety of application domains. The maximum \(\gamma \)-quasi-clique problem is to find a \(\gamma \)-quasi-clique of maximum cardinality in the graph and is known to be NP-hard. This paper proposes new mixed integer programming (MIP) formulations for solving the maximum \(\gamma \)-quasi-clique problem. The corresponding linear programming (LP) relaxations are analyzed and shown to be tighter than the LP relaxations of the MIP models available in the literature on sparse graphs. The developed methodology is naturally generalized for solving the maximum \(f(\cdot )\)-dense subgraph problem, which, for a given function \(f(\cdot )\), seeks for the largest k such that there is a subgraph induced by k vertices with at least f(k) edges. The performance of the proposed exact approaches is illustrated on real-life network instances with up to 10,000 vertices.

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

References

  1. Abello, J., Pardalos, P.M., Resende, M.G.C.: On maximum clique problems in very large graphs. In: Abello, J., Vitter, J. (eds.) External Memory Algorithms and Visualization, pp. 119–130. American Mathematical Society, Boston (1999)

    Google Scholar 

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

    Chapter  Google Scholar 

  3. Adams, W.P., Forrester, R.J., Glover, F.W.: Comparisons and enhancement strategies for linearizing mixed 0–1 quadratic programs. Discret. Optim. 1(2), 99–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Asahiro, Y., Hassin, R., Iwama, K.: Complexity of finding dense subgraphs. Discret. Appl. Math. 121(1–3), 15–26 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bader, G.D., Hogue, C.W.: An automated method for finding molecular complexes in large protein interaction networks. BMC Bioinf. 4(1), 2 (2003)

    Article  Google Scholar 

  6. Balasundaram, B., Butenko, S., Hicks, I.: Clique relaxations in social network analysis: the maximum \(k\)-plex problem. Oper. Res. 59(1), 133–142 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Batagelj, V., Mrvar, A.: Pajek Datasets. http://vlado.fmf.uni-lj.si/pub/networks/data/ (2006). Accessed 26 Oct 2015

  9. Bhattacharyya, M., Bandyopadhyay, S.: Mining the largest quasi-clique in human protein interactome. In: Proceedings of the 2009 International Conference on Adaptive and Intelligent Systems, ICAIS ’09, pp. 194–199. IEEE Computer Society, Washington, DC, USA (2009)

  10. Boginski, V., Butenko, S., Pardalos, P.: Mining market data: a network approach. Comput. Oper. Res. 33(11), 3171–3184 (2006)

    Article  MATH  Google Scholar 

  11. Boginski, V., Butenko, S., Pardalos, P.M.: Statistical analysis of financial networks. Comput. Stat. Data Anal. 48(2), 431–443 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. Handbook of Combinatorial Optimization, vol. 4, pp. 1–74. Kluwer Academic Publishers, Boston (1999)

    Chapter  Google Scholar 

  13. Bu, D., Zhao, Y., Cai, L., Xue, H., Zhu, X., Lu, H., Zhang, J., Sun, S., Ling, L., Zhang, N., Li, G., Chen, R.: Topological structure analysis of the protein–protein interaction network in budding yeast. Nucleic Acids Res. 31(9), 2443–2450 (2003)

    Article  Google Scholar 

  14. Butenko, S., Wilhelm, W.: Clique-detection models in computational biochemistry and genomics. Eur. J. Oper. Res. 173(1), 1–17 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chung, F., Lu, L., Vu, V.: The spectra of random graphs with given expected degrees. Internet Math. 1, 257–275 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Crenson, M.A.: Social networks and political processes in urban neighborhoods. Am. J. Polit. Sci. 22(3), 578–594 (1978)

    Article  Google Scholar 

  17. Davis, S., Trapman, P., Leirs, H., Begon, M., Heesterbeek, J.A.P.: The abundance threshold for plague as a critical percolation phenomenon. Nature 454, 634–637 (2008)

    Article  Google Scholar 

  18. Davis, T.A., Hu, Y.: The University of Florida sparse matrix collection. ACM Trans. Math. Softw. 38(1), 1–25 (2011)

    MathSciNet  Google Scholar 

  19. Erdős, P., Rényi, A.: On random graphs. Publicationes Mathematicae Debrecen 6, 290–297 (1959)

    MathSciNet  MATH  Google Scholar 

  20. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H Freeman, New York (1979)

    MATH  Google Scholar 

  21. Guo, J., Kanj, I.A., Komusiewicz, C., Uhlmann, J.: Editing graphs into disjoint unions of dense clusters. Algorithmica 61(4), 949–970 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1952)

    MATH  Google Scholar 

  23. Hartwell, L.H., Hopfield, J.J., Leibler, S., Murray, A.W.: From molecular to modular cell biology. Nature 402(6761 Suppl), C47–C52 (1999)

    Article  Google Scholar 

  24. Holzapfel, K., Kosub, S., Maaß, M.G., Täubig, H.: The complexity of detecting fixed-density clusters. Algorithms and Complexity, pp. 201–212. Springer, Berlin (2003)

    Chapter  Google Scholar 

  25. Hu, H., Yan, X., Huang, Y., Han, J., Zhou, X.J.: Mining coherent dense subgraphs across massive biological networks for functional discovery. Bioinformatics 21(suppl 1), i213–i221 (2005)

    Article  Google Scholar 

  26. Huang, W.-Q., Zhuang, X.-T., Yao, S.: A network analysis of the Chinese stock market. Phys. A: Stat. Mech. Appl. 388(14), 2956–2964 (2009)

    Article  Google Scholar 

  27. Ibaraki, T.: Integer programming formulation of combinatorial optimization problems. Discret. Math. 16(1), 39–52 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kleinberg, J., Goemans, M.X.: The Lovász theta function and a semidefinite programming relaxation of vertex cover. SIAM J. Discret. Math. 11(2), 196–204 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25(1), 1–7 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  30. Luce, R.D., Perry, A.D.: A method of matrix analysis of group structure. Psychometrika 14(2), 95–116 (1949)

    Article  MathSciNet  Google Scholar 

  31. Matsuda, H., Ishihara, T., Hashimoto, A.: Classifying molecular sequences using a linkage graph with their pairwise similarities. Theor. Comput. Sci. 210, 305–325 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)

    Book  MATH  Google Scholar 

  33. Östergård, P.: A fast algorithm for the maximum clique problem. Discret. Appl. Math. 120(1), 197–207 (2002)

    Article  MathSciNet  Google Scholar 

  34. Pajouh, F.M., Miao, Z., Balasundaram, B.: A branch-and-bound approach for maximum quasi-cliques. Ann. Oper. Re. 216(1), 145–161 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pardalos, P.M., Xue, J.: The maximum clique problem. J. Global. Optim. 4(3), 301–328 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pattillo, J., Veremyev, A., Butenko, S., Boginski, V.: On the maximum quasi-clique problem. Discret. Appl. Math. 161(1–2), 244–257 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pattillo, J., Youssef, N., Butenko, S.: On clique relaxation models in network analysis. Eur. J. Oper. Res. 226(1), 9–18 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ravasz, E., Somera, A.L., Mongru, D.A., Oltvai, Z.N., Barabási, A.L.: Hierarchical organization of modularity in metabolic networks. Science 297(5586), 1551–1555 (2002)

    Article  Google Scholar 

  39. Saban, D., Bonomo, F., Stier-Moses, N.E.: Analysis and models of bilateral investment treaties using a social networks approach. Phys. A: Stat. Mech. Appl. 389(17), 3661–3673 (2010)

    Article  Google Scholar 

  40. Scott, J.: Social Network Analysis: A Handbook, 2nd edn. Sage Publications, London (2000)

    Google Scholar 

  41. Sim, K., Li, J., Gopalkrishnan, V., Liu, G.: Mining maximal quasi-bicliques to co-cluster stocks and financial ratios for value investment. In: Proceedings of the Sixth International Conference on Data Mining, ICDM ’06, pp. 1059–1063. IEEE Computer Society, Washington, DC, USA (2006)

  42. Spirin, V., Mirny, L.A.: Protein complexes and functional modules in molecular networks. Proc. Natl. Acad. Sci. 100(21), 12123–12128 (2003)

    Article  Google Scholar 

  43. Trick, M.: COLOR02/03/04: Graph Coloring and Its Generalizations. http://mat.gsia.cmu.edu/COLOR03/ (2004). Accessed 26 Oct 2015

  44. Trukhanov, S., Balasubramaniam, C., Balasundaram, B., Butenko, S.: Algorithms for detecting optimal hereditary structures in graphs, with application to clique relaxations. Comput. Optim. Appl. 56(1), 113–130 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Uno, T.: An efficient algorithm for solving pseudo clique enumeration problem. Algorithmica 56, 3–16 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Veremyev, A., Boginski, V., Krokhmal, P.A., Jeffcoat, D.E.: Dense percolation in large-scale mean-field random networks is provably “explosive”. PLoS One 7(12), e51883 (2012)

    Article  Google Scholar 

  47. Vogiatzis, C., Veremyev, A., Pasiliao, E.L., Pardalos, P.M.: An integer programming approach for finding the most and the least central cliques. Optim. Lett. 9(4), 615–633 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Wasserman, S., Faust, K.: Social Network Analysis. Cambridge University Press, New York (1994)

    Book  MATH  Google Scholar 

  49. FICO\(^{\text{ TM }}\) Xpress Optimization Suite 7.6. http://www.fico.com/en/Products/DMTools/Pages/FICO-Xpress-Optimization-Suite.aspx (2014). Accessed 26 Oct 2015

  50. Yu, H., Paccanaro, A., Trifonov, V., Gerstein, M.: Predicting interactions in protein networks by completing defective cliques. Bioinformatics 22(7), 823–829 (2006)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments that helped us to greatly improve the quality of the paper. This material is based upon work supported by the U.S. Air Force Research Laboratory (AFRL) Mathematical Modeling and Optimization Institute and the U.S. Air Force Office of Scientific Research (AFOSR). The research of the first author was performed while he held a National Research Council Research Associateship Award at AFRL Munitions Directorate. The research of the second author was also supported by the U.S. Air Force Summer Faculty Fellowship and by AFRL/RW under agreement number FA8651-14-2-0002. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of AFRL/RW or the U.S. Government.

Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Alexander Veremyev

  2. Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Oleg A. Prokopyev

  3. Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX, 77843, USA

    Sergiy Butenko

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

    Eduardo L. Pasiliao

Authors
  1. Alexander Veremyev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oleg A. Prokopyev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sergiy Butenko
    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 Sergiy Butenko.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Veremyev, A., Prokopyev, O.A., Butenko, S. et al. Exact MIP-based approaches for finding maximum quasi-cliques and dense subgraphs. Comput Optim Appl 64, 177–214 (2016). https://doi.org/10.1007/s10589-015-9804-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-015-9804-y

Keywords

Search

Navigation