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Integer programming formulations and efficient local search for relaxed correlation clustering

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Abstract

Relaxed correlation clustering (RCC) is a vertex partitioning problem that aims at minimizing the so-called relaxed imbalance in signed graphs. RCC is considered to be an NP-hard unsupervised learning problem with applications in biology, economy, image recognition and social network analysis. In order to solve it, we propose two linear integer programming formulations and a local search-based metaheuristic. The latter relies on auxiliary data structures to efficiently perform move evaluations during the search process. Extensive computational experiments on existing and newly proposed benchmark instances demonstrate the superior performance of the proposed approaches when compared to those available in the literature. While the exact approaches obtained optimal solutions for open problems, the proposed heuristic algorithm was capable of finding high quality solutions within a reasonable CPU time. In addition, we also report improving results for the symmetrical version of the problem. Moreover, we show the benefits of implementing the efficient move evaluation procedure that enables the proposed metaheuristic to be scalable, even for large-size instances.

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References

  1. Ales, Z., Knippel, A., Pauchet, A.: Polyhedral combinatorics of the \(k\)-partitioning problem with representative variables. Discret. Appl. Math. 211, 1–14 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Altafini, C.: Dynamics of opinion forming in structurally balanced social networks. PLoS ONE 7(6), 1–9 (2012)

    Article  Google Scholar 

  3. Arinik, N., Figueiredo, R., Labatut, V.: Signed Graph Analysis for the Interpretation of Voting Behavior. In: International Conference on Knowledge Technologies and Data-driven Business (i-KNOW), Graz, Austria, International Workshop on Social Network Analysis and Digital Humanities (SnanDig) (2017)

  4. Bahiense, L., Frota, Y., Maculan, N., Noronha, T.F., Ribeiro, C.C.: A branch-and-cut algorithm for equitable coloring based on a formulation by representatives. Electron. Notes Discret. Math. 35, 347–352 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56(1), 89–113 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beier, T., Hamprecht, F.A., Kappes, J.H.: Fusion moves for correlation clustering. In: CVPR. Proceedings, pp 3507–3516, 1 (2015)

  7. Bonami, P., Nguyen, V.H., Klein, M., Minoux, M.: On the solution of a graph partitioning problem under capacity constraints. In: Mahjoub, A.R., Markakis, V., Milis, I., Paschos, V.T. (eds.) Combinatorial Optimization, pp. 285–296. Springer, Berlin (2012)

    Chapter  Google Scholar 

  8. Broder, A.Z.: The r-stirling numbers. Discret. Math. 49(3), 241–259 (1984)

    Article  MATH  Google Scholar 

  9. Brusco, M., Doreian, P., Mrvar, A., Steinley, D.: Two algorithms for relaxed structural balance partitioning: linking theory, models, and data to understand social network phenomena. Sociol. Methods Res. 40(1), 57–87 (2011)

    Article  MathSciNet  Google Scholar 

  10. Brusco, M.J., Doreian, P.: Partitioning signed networks using relocation heuristics, tabu search, and variable neighborhood search. Soc. Netw. 56, 70–80 (2019)

    Article  Google Scholar 

  11. Bulhões, T., de Sousa Filho, G.F., Subramanian, A., Lucídio dos Anjos, F.C.: Branch-and-cut approaches for p-cluster editing. Discret. Appl. Math. 219, 51–64 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Campêlo, M., Campos, V.A., Correa, R.C.: On the asymmetric representatives formulation for the vertex coloring problem. Discret. Appl. Math. 156, 1097–1111 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Campêlo, M.B., Corrêa, R.C., Frota, Y.: Cliques, holes and the vertex coloring polytope. Inf. Process. Lett. 89(4), 159–164 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cartwright, D., Harary, F.: Structural balance: a generalization of heider’s theory. Psychol. Rev. 63(5), 277 (1956)

    Article  Google Scholar 

  15. Dambacher, J.M., Li, H.W., Rossignol, P.A.: Relevance of community structure in assessing indeterminacy of ecological predictions. Ecology 83(5), 1372–1385 (2002)

    Article  Google Scholar 

  16. DasGupta, B., AEnciso, G., Sontag, E., Zhang, Y.: Algorithmic and complexity results for decompositions of biological networks into monotone subsystems. Bio. Syst. 90, 161–178 (2007)

    Google Scholar 

  17. Davis, J.A.: Clustering and structural balance in graphs. Hum. Relat. 20(2), 181–187 (1967)

    Article  Google Scholar 

  18. Doreian, P.: A multiple indicator approach to blockmodeling signed networks. Soc. Netw. 30(3), 247–258 (2008)

    Article  MathSciNet  Google Scholar 

  19. Doreian, P., Mrvar, A.: A partitioning approach to structural balance. Soc. Netw. 18(2), 149–168 (1996)

    Article  Google Scholar 

  20. Doreian, P., Mrvar, A.: Partitioning signed social networks. Soc. Netw. 31(1), 1–11 (2009)

    Article  MATH  Google Scholar 

  21. Doreian, P., Mrvar, A.: Testing two theories for generating signed networks using real data (2014)

  22. Doreian, P., Mrvar, A.: Structural balance and signed international relations. J. Soc. Struct. 16, 2 (2015)

    Article  Google Scholar 

  23. Facchetti, G., Iacono, G., Altafini, C.: Computing global structural balance in large-scale signed social networks. Proc. Natl. Acad. Sci. 108(52), 20953–20958 (2011)

    Article  Google Scholar 

  24. Fan, N., Pardalos, P.M.: Linear and quadratic programming approaches for the general graph partitioning problem. J. Glob. Optim. 48(1), 57–71 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Figueiredo, R., Frota, Y.: The maximum balanced subgraph of a signed graph: applications and solution approaches. Eur. J. Oper. Res. 236(2), 473–487 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Figueiredo, R., Moura, G.: Mixed integer programming formulations for clustering problems related to structural balance. Soc. Netw. 35(4), 639–651 (2013)

    Article  Google Scholar 

  27. Figueiredo, R., Frota, Y., Labbé, M.: A branch-and-cut algorithm for the maximum k-balanced subgraph of a signed graph. Discret. Appl. Math. 2, 5 (2018)

    MATH  Google Scholar 

  28. Frota, Y., Maculan, N., Noronha, T.F., Ribeiro, C.C.: A branch-and-cut algorithm for partition coloring. Network 55, 194–204 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Harary, F., Lim, M., Wunsch, D.C.: Signed graphs for portfolio analysis in risk management. IMA J. Manag. Math. 13, 1–10 (2003)

    Google Scholar 

  30. Heider, F.: Attitudes and cognitive organization. J. Psychol. 21(1), 107–112 (1946). pMID: 21010780

    Article  Google Scholar 

  31. Kim, S., Nowozin, S., Kohli, P., Yoo, C.D.: Higher-order correlation clustering for image segmentation. In: Shawe-Taylor, J., Zemel, R.S., Bartlett, P.L., Pereira, F., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, vol. 24, pp. 1530–1538. Current Associates Inc., New York (2011)

    Google Scholar 

  32. Lemann, T.B., Solomon, R.L.: Group characteristics as revealed in sociometric patterns and personality ratings. Sociom 15(1/2), 7–90 (1952)

    Article  Google Scholar 

  33. Levorato, M.: Efficient solutions to the correlation clustering problem. Master’s thesis, Universidade Federal Fluminense, Niterói, Rio de Janeiro, Brazil, (2015). http://www.ic.uff.br/PosGraduacao/frontend-tesesdissertacoes/download.php?id=700.pdf&tipo=trabalho

  34. Levorato, M., Frota, Y.: Brazilian congress structural balance analysis. J. Interdiscip. Methodol. Issues Sci. 3, 10 (2017)

    Google Scholar 

  35. Levorato, M., Drummond, L., Frota, Y., Figueiredo, R.: An ils algorithm to evaluate structural balance in signed social networks. In: Proceedings of the 30th Annual ACM Symposium on Applied Computing, ACM, New York, SAC ’15, pp 1117–1122 (2015)

  36. Levorato, M., Figueiredo, R., Frota, Y., Drummond, L.: Evaluating balancing on social networks through the efficient solution of correlation clustering problems. EURO J. Comput. Optim. 5(4), 467–498 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lourenço, H.R., Martin, O.C., Stützle, T.: Iterated local search: Framework and applications. In: Gendreau, M., Potvin, J.Y. (eds.) Handbook of Metaheuristics, pp. 363–397. Springer, Boston (2010)

    Chapter  Google Scholar 

  38. Maurya, M.R., Rengaswamy, R., Venkatasubramanian, V.: Application of signed digraphs-based analysis for fault diagnosis of chemical process flowsheets. Eng. Appl. Artif. Intell. 17(5), 501–518 (2004)

    Article  Google Scholar 

  39. McKinney, J.C.: An educational application of a two-dimensional sociometric test. Sociom 11(4), 356–367 (1948)

    Article  Google Scholar 

  40. Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  41. Newcomb, T.M.: The Acquaintance Process. Holt, Rinehart & Winston, New York (1961)

    Book  Google Scholar 

  42. Sampson, S.F.: A novitiate in a period of change: an experimental and case study of social relationships. Ph.D. thesis, Department of Sociology, Cornell University, NY (1968)

  43. Silva, M.M., Subramanian, A., Vidal, T., Ochi, L.S.: A simple and effective metaheuristic for the minimum latency problem. Eur. J. Oper. Res. 221(3), 513–520 (2012)

    Article  MATH  Google Scholar 

  44. Silva, M.M., Subramanian, A., Ochi, L.S.: An iterated local search heuristic for the split delivery vehicle routing problem. Comput. Oper. Res. 53, 234–249 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Subramanian, A., Farias, K.: Efficient local search limitation strategy for single machine total weighted tardiness scheduling with sequence-dependent setup times. Comput. Oper. Res. 79, 190–206 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  46. Van Gael, J., Zhu, X.: Correlation clustering for crosslingual link detection. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, IJCAI’07, pp. 1744–1749 (2007)

  47. Vasanthi, B., Arumugam, S., Nagar, A.K., Mitra, S.: Applications of signed graphs to portfolio turnover analysis. In: 2nd Global Conference on Business and Social Sciences (GCBSS-2015) on “Multidisciplinary Perspectives on Management and Society”, 2015, Bali, Indonesia (2015)

  48. Wang, N., Li, J.: Restoring: a greedy heuristic approach based on neighborhood for correlation clustering. In: Motoda, H., Wu, Z., Cao, L., Zaiane, O., Yao, M., Wang, W. (eds.) Advanced Data Mining and Applications, pp. 348–359. Springer, Berlin (2013)

    Chapter  Google Scholar 

  49. Yang, B., Cheung, W., Liu, J.: Community mining from signed social networks. IEEE Trans. Knowl. Data Eng. 19, 1333–1348 (2007)

    Article  Google Scholar 

  50. Zaslavsky, T.: Signed graphs. Discret. Appl. Math. 4, 47–74 (1982)

    Article  MATH  Google Scholar 

  51. Zaslavsky, T.: A mathematical bibliography of signed and gain graphs and allied areas. Electron. J. Comb. DS 3, 8 (1998)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior–Brasil (CAPES)–Finance Code 001 and by the Conselho Nacional de Desenvolvimento Cientfífico e Tecnológico (CNPq), grants 305223/2015-1 and 303799/2018-8. We would also like to thank Mário Levorato for providing the SRCC source code and Pedro Liguori for the helpful insights on the mathematical models.

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Authors and Affiliations

  1. Instituto de Computação, Universidade Federal Fluminense, Rua Passo da Pátria, São Domingos, 24210-240, Niterói-RJ, Brazil

    Eduardo Queiroga & Yuri Frota

  2. Departamento de Sistemas de Computação, Centro de Informática, Universidade Federal da Paraíba, Rua dos Escoteiros, Mangabeira, 58055-000, João Pessoa, Brazil

    Anand Subramanian

  3. Laboratoire Informatique d’Avignon, Avignon Université, 339 Chemin des Meinajaries, 84911, Avignon cedex 9, France

    Rosa Figueiredo

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  1. Eduardo Queiroga
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Correspondence to Eduardo Queiroga.

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Appendices

Appendix: Detailed results for the random RCC instances

Table 13 Relaxed imbalance obtained by ILS\(_\text {RCC}\) and ILS\(_\text {adapt}\)
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Table 14 Best solution of all experiments
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Detailed results for the SRCC instances

Table 15 Symmetric relaxed imbalance obtained by ILS\(_\text {RCC}\) and ILS Levorato et al. [36]
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Queiroga, E., Subramanian, A., Figueiredo, R. et al. Integer programming formulations and efficient local search for relaxed correlation clustering. J Glob Optim 81, 919–966 (2021). https://doi.org/10.1007/s10898-020-00989-7

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