Skip to main content
SpringerLink
Log in

Branch-and-price for p-cluster editing

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

Abstract

Given an input graph, the p-cluster editing problem consists of minimizing the number of editions, i.e., additions and/or deletions of edges, so as to create p vertex-disjoint cliques (clusters). In order to solve this \({\mathscr {NP}}\)-hard problem, we propose a branch-and-price algorithm over a set partitioning based formulation with exponential number of variables. We show that this formulation theoretically dominates the best known formulation for the problem. Moreover, we compare the performance of three mathematical formulations for the pricing subproblem, which is strongly \({\mathscr {NP}}\)-hard. A heuristic algorithm is also proposed to speedup the column generation procedure. We report improved bounds for benchmark instances available in the literature.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: ranking and clustering. J. ACM 55(5), 23:1–23:27 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alidaee, B., Glover, F., Kochenberger, G., Wang, H.: Solving the maximum edge weight clique problem via unconstrained quadratic programming. Eur. J. Oper. Res. 181(2), 592–597 (2007)

    Article  MATH  Google Scholar 

  3. Alizadeh, A.A., Eisen, M.B., Davis, R.E., Ma, C., Lossos, I.S., Rosenwald, A., Boldrick, J.C., Sabet, H., Tran, T., Yu, X., Powell, J.I., Yang, L., Marti, G.E., Moore, T., Hudson, J., Lu, L., Lewis, D.B., Tibshirani, R., Sherlock, G., Chan, W.C., Greiner, T.C., Weisenburger, D.D., Armitage, J.O., Warnke, R., Levy, R., Wilson, W., Grever, M.R., Byrd, J.C., Botstein, D., Brown, P.O., Staudt, L.M.: Distinct types of diffuse large B-cell lymphoma identified by gene expression profiling. Nature 403(6769), 503–511 (2000)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Barnhart, C., Johnson, E.L., Nemhauser, G.L., Savelsbergh, M.W.P., Vance, P.H.: Branch-and-price: column generation for solving huge integer programs. Oper. Res. 46, 316–329 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bastos, L., Ochi, L.S., Protti, F., Subramanian, A., Martins, I.C., Pinheiro, R.: Efficient algorithms for cluster editing. J. Comb. Optim. 31(1), 347–371 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben-Dor, A., Shamir, R., Yakhini, Z.: Clustering gene expression patterns. J. Comput. Biol. 6(3–4), 281–297 (1999)

    Article  Google Scholar 

  8. Billionnet, A., Soutif, E.: Using a mixed integer programming tool for solving the 0–1 quadratic knapsack problem. INFORMS J. Comput. 16(2), 188–197 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Böcker, S.: A golden ratio parameterized algorithm for cluster editing. In: Iliopoulos C., Smyth W. (eds.) Combinatorial Algorithms. Lecture Notes in Computer Science, vol. 7056, pp. 85–95. Springer,Berlin (2011)

  10. Böcker, S., Baumbach, J.: Cluster editing. In: Bonizzoni P., Brattka V., Löwe B. (eds.) The Nature of Computation. Logic, Algorithms, Applications. Lecture Notes in Computer Science, vol. 7921, pp.33–44. Springer, Berlin (2013)

  11. Böcker, S., Briesemeister, S., Bui, Q., Truss, A.: Goingweighted: parameterized algorithms for cluster editing. In: Yang B., Du D.Z., Wang C. (eds.) CombinatorialOptimization and Applications.Lecture Notes in Computer Science, vol. 5165, pp. 1–12. Springer, Berlin (2008)

  12. Böcker, S., Briesemeister, S., Klau, G.: Exact algorithms for cluster editing: evaluation and experiments. Algorithmica 60(2), 316–334 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Böcker, S., Damaschke, P.: Even faster parameterized cluster deletion and cluster editing. Inf. Process. Lett. 111(14), 717–721 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bulhões, T., Subramanian, A., Sousa Filho, G.F., Cabral, L.A.F.:Branch-and-cut approaches for p-cluster editing. Discrete Appl. Math. (2016). doi:10.1016/j.dam.2016.10.026 (to appear)

  15. Charikar, M., Guruswami, V., Wirth, A.: Clustering with qualitative information. J. Comput. Syst. Sci. 71(3), 360–383 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, D.S., Batson, R.G., Dang, Y.: Applied Integer Programming. Wiley, New York (2009)

    Book  MATH  Google Scholar 

  17. de Henrique Paiva Perché, M.: Metaheurísticas híbridas aplicadas ao problema de edição não automática de clusters. Master’s thesis, Universidade Federal Fluminense - UFF, Brasil (2012) (in Portuguese)

  18. Dehne, F., Langston, M.A., Luo, X., Pitre, S., Shaw, P., Zhang, Y.: The cluster editing problem: implementations and experiments. Lect. Notes Comput. Sci. 4169, 13–24 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fomin, F.V., Kratsch, S., Pilipczuk, M., Pilipczuk, M., Villanger, Y.: Tight bounds for parameterized complexity of cluster editing with a small number of clusters. J. Comput. Syst. Sci. 80(7), 1430–1447 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Giotis, I., Guruswami, V.: Correlation clustering with a fixednumber of clusters. In: Proceedings of the Seventeenth AnnualACM-SIAM Symposium on Discrete Algorithm, SODA ’06, pp. 1167–1176.ACM, New York, NY (2006)

  21. Golub, T.R., Slonim, D.K., Tamayo, P., Huard, C., Gaasenbeek, M., Mesirov, J.P., Coller, H., Loh, M.L., Downing, J.R., Caligiuri, M.A., Bloomfield, C.D., Lander, E.S.: Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science 286(5439), 531–537 (1999)

    Article  Google Scholar 

  22. Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeleddata clustering: fixed-parameter algorithms for clique generation. In: Petreschi R., Persiano G., Silvestri R. (eds.) Algorithms and Complexity. Lecture Notes in Computer Science, vol. 2653, pp. 108–119. Springer, Berlin (2003)

  23. Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39(4), 321–347 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Grötschel, M., Wakabayashi, Y.: A cutting plane algorithm for a clustering problem. Math. Program. 45(1), 59–96 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  25. Guo, J.: A more effective linear kernelization for cluster editing. In: Chen B., Paterson M., Zhang G. (eds.) Combinatorics, Algorithms, Probabilistic and Experimental Methodologies. Lecture Notes in Computer Science, vol. 4614, pp. 36–47. Springer, Berlin (2007)

  26. Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J., Bohlinger, J. (eds.) Complexity of Computer Computations, The IBM Research Symposia Series, pp. 85–103. Springer, New York (1972)

    Chapter  Google Scholar 

  27. Komusiewicz, C., Uhlmann, J.: Cluster editing with locally bounded modifications. Discrete Appl. Math. 160(15), 2259–2270 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Pisinger, D.: The quadratic knapsack problem—a survey. Discrete Appl. Math. 155(5), 623–648 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Protti, F., Silva, M.D., Szwarcfiter, J.: Applying modular decomposition to parameterized cluster editing problems. Theor. Comput. Syst. 44, 91–104 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rahmann, S., Wittkop, T., Baumbach, J., Martin, M., Truss, A., Böcker, S.: Exact and heuristic algorithms for weighted cluster editing. In: Markstein P., Xu Y. (eds.) Computational Systems Bioinformatics: CSB 2007 Conference Proceedings, vol. 6, pp. 391–400. Imp. Coll. Press, 57 Shelton Street, Covent Garden, London WC2H 9HE (2007)

  32. Ryan, D.M., Foster, B.A.: Computer scheduling of public transport: urban passenger vehicle and crew scheduling, chap. An integer programming approach to scheduling, pp. 269–280. North-Holland, Amsterdam (1981)

  33. Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. In: Goos G., Hartmanis J., Leeuwen J., Kučcera L. (eds.) Graph-Theoretic Concepts in Computer Science. Lecture Notes in Computer Science, vol. 2573, pp. 379–390. Springer, Berlin (2002)

  34. Sörensen, M.M.: New facets and a branch-and-cut algorithm for the weighted clique problem. Eur. J. Oper. Res. 154(1), 57–70 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Grant 305223/2015-1.

Author information

Authors and Affiliations

  1. Instituto de Computação, Universidade Federal Fluminense, Rua Passo da Pátria 156, São Domingos, Niterói, RJ, 24210-240, Brazil

    Teobaldo Bulhões

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

    Anand Subramanian

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

    Gilberto F. Sousa Filho & Lucídio dos Anjos F. Cabral

Authors
  1. Teobaldo Bulhões
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anand Subramanian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gilberto F. Sousa Filho
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lucídio dos Anjos F. Cabral
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anand Subramanian.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 91 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bulhões, T., Subramanian, A., Sousa Filho, G.F. et al. Branch-and-price for p-cluster editing. Comput Optim Appl 67, 293–316 (2017). https://doi.org/10.1007/s10589-017-9893-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-017-9893-x

Keywords

Search

Navigation