This paper presents a hybrid Constraint Programming (CP) and Semidefinite Programming (SDP) approach to the k-clustering minimum biclique completion problem on bipartite graphs. The problem consists in partitioning a bipartite undirected graph into k clusters such that the sum of the edges that complete each cluster into a biclique, i.e., a complete bipartite subgraph, is minimum. The problem arises in telecommunications, in particular in bundling channels in multicast transmissions. In literature, the problem has been tackled with an Integer Bilinear Programming approach. We introduce two quasi-biclique constraints and we propose a SDP relaxation of the problem that provides much stronger lower bounds than the Bilinear Programming relaxation. The quasi-biclique constraints and the SDP relaxation are integrated into a hybrid CP and SDP approach. Computational results on a set of random instances provide further evidence about the potential of CP and SDP hybridizations.


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  1. 1.
    Monson, S., Pullman, N., Rees, R.: A survey of clique and biclique coverings and factorizations of (0,1)-matrices. Bull. of the Combin. and its Appl. 14, 17–86 (1992)MathSciNetMATHGoogle Scholar
  2. 2.
    Faure, N., Chrétienne, P., Gourdin, E., Sourd, F.: Biclique completion problems for multicast network design. Discrete Optim. 4(3), 360–377 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Fahle, T.: Simple and fast: Improving a branch-and-bound algorithm for maximum clique. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 485–498. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Régin, J.C.: Using constraint programming to solve the maximum clique problem. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 634–648. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Barnier, N., Brisset, P.: Graph coloring for air traffic flow management. Ann. of Oper. Res. 130, 163–178 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Focacci, F., Lodi, A., Milano, M.: Cost-based domain filtering. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 189–203. Springer, Heidelberg (1999)Google Scholar
  7. 7.
    van Hoeve, W.: Exploiting semidefinite relaxations in constraint programming. Computers & OR 33, 2787–2804 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gomes, C., van Hoeve, W.J., Leahu, L.: The power of semidefinite programming relaxations for MAXSAT. In: Beck, J.C., Smith, B.M. (eds.) CPAIOR 2006. LNCS, vol. 3990, pp. 104–118. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. of the ACM 42, 1115–1145 (1995)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Frieze, A., Jerrum, M.: Improved approximation algorithms for max k-cut and max-bisection. Algorithmica 18, 67–81 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Régin, J.C.: Global Constraints and Filtering Algorithms. In: Milano, M. (ed.) Constraint and Integer Programming-Toward a Unified Methodology. Kluwer, Dordrecht (2004)Google Scholar
  12. 12.
    Gecode: Generic constraint development environment, http://www.gecode.org
  13. 13.
    Benson, S.J., Ye, Y.: Algorithm 875: DSDP5—software for semidefinite programming. ACM Trans. Math. Softw. 34(3), 1–20 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Stefano Gualandi
    • 1
  1. 1.Dipartimento di Elettronica e InformazionePolitecnico di MilanoItaly

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