Optimization Letters

, Volume 11, Issue 1, pp 47–54 | Cite as

A new family of facet defining inequalities for the maximum edge-weighted clique problem

Original Paper
First Online:
Received:
Accepted:
  • 140 Downloads

Abstract

This paper considers a family of cutting planes, recently developed for mixed 0–1 polynomial programs and shows that they define facets for the maximum edge-weighted clique problem. There exists a polynomial time exact separation algorithm for these inequalities. The result of this paper may contribute to the development of more efficient algorithms for the maximum edge-weighted clique problem that use cutting planes.

Keywords

Edge-weighted clique problem Cutting planes Separation algorithm Integer programming Boolean quadric polytope Facet defining inequalities 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

Part of the work of the author was done during a stay at GERAD as a postdoctoral researcher with Professor Miguel Anjos. The author is grateful to Professor Adam Letchford for his comments and feedback on this work, and finally thanks the referees for their careful reading of the paper and their helpful feedback.

References

  1. 1.
    Alidaee, B., Glover, F., Kochenberger, G., Wang, H.: Solving the maximum edge weight clique problem via unconstrained quadratic programming. Eur. J. Oper. Res. 181, 592–597 (2007)CrossRefMATHGoogle Scholar
  2. 2.
    Caprara, A., Pisinger, D., Toth, P.: Exact solution of the quadratic knapsack problem. INFORMS J. Comput. 11, 125–137 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dijkhuizen, G., Faigle, U.: A cutting-plane approach to the edge-weighted maximal clique problem. Eur. J. Oper. Res. 69(1), 121–130 (1993)CrossRefMATHGoogle Scholar
  4. 4.
    Djeumou Fomeni, F., Kaparis, K., Letchford, A.N.: Cutting planes for first-level RLT relaxations of mixed 0–1 programs. Math. Progr. 151(2), 639–658 (2015)CrossRefMATHGoogle Scholar
  5. 5.
    Djeumou Fomeni, F., Letchford, A.N.: A dynamic programming heuristic for the quadratic knapsack problem. INFORMS J. Comp. 26(1), 173–183 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hunting, M.: Relaxation techniques for discrete optimization problems: theory and algorithms. PhD thesis, University of Twente (1998)Google Scholar
  7. 7.
    Hunting, M., Faigle, U., Kern, W.: A Lagrangean relaxation approach to the edge-weighted clique problem. Eur. J. Oper. Res. 131, 119–131 (2001)CrossRefMATHGoogle Scholar
  8. 8.
    Helmberg, C., Rendl, F., Weismantel, R.: A semidefinite programming approach to the quadratic knapsack problem. J. Comb. Optim. 4, 197–215 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Johnson, E.L., Mehrotra, A., Nemhauser, G.L.: Min-cut clustering. Math. Progr. 62, 133–151 (1993)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kuby, M.J.: Programming models for facility dispersion: the \(p\)-dispersion and maxisum dispersion problem. Geograph. Anal. 9, 315–329 (1987)Google Scholar
  11. 11.
    Macambria, E.M., de Souza, C.C.: The edge-weight clique problem: valid inequalities, facets and polyhedral computations. Eur. J. Oper. Res. 123, 346–371 (2000)CrossRefGoogle Scholar
  12. 12.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I-convex underestimating problems. Math. Progr. 10, 146–175 (1979)Google Scholar
  13. 13.
    Mehrotra, A.: Cardinality constrained boolean quadratic polytope. Disc. Appl. Math 79, 137–154 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Park, K., Lee, K., Park, S.: An extended formulation approach to the edge-weight maximal clique problem. Eur. J. Oper. Res. 95, 671–682 (1995)CrossRefMATHGoogle Scholar
  15. 15.
    Padberg, M.W.: The Boolean quadric polytope: some characteristics, facets and relatives. Math. Progr. 45, 139–172 (1989)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pisinger, D.: The quadratic knapsack problem: a survey. Discr. Appl. Math. 155, 623–648 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ravi, S.S., Rosenkrantz, D.J., Tayi, G.K.: Heuristic and special case algorithms for dispersion problems. Oper. Res. 42(2), 299–310 (1994)CrossRefMATHGoogle Scholar
  18. 18.
    Späth, H.: Heuristically determining cliques of given cardinality and with minimal cost within weighted complete graphs. Z. Oper. Res. 29(3), 125–131 (1985)MathSciNetMATHGoogle Scholar
  19. 19.
    Sørensen, M.M.: New facets and a branch-and-cut algorithm for the weighted clique problem. Eur. J. Oper. Res. 154, 57–70 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wu, Q., Hao, J.K.: A review on algorithms for maximum clique problems. Eur. J. Oper. Res. 242(3), 693–709 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.GERAD and Department of Mathematics and Industrial EngineeringPolytechnique MontréalMontrealCanada

Personalised recommendations