Journal of Combinatorial Optimization

, Volume 32, Issue 2, pp 531–549 | Cite as

Solving the maximum vertex weight clique problem via binary quadratic programming

  • Yang Wang
  • Jin-Kao Hao
  • Fred Glover
  • Zhipeng Lü
  • Qinghua Wu
Article

Abstract

In recent years, the general binary quadratic programming (BQP) model has been widely applied to solve a number of combinatorial optimization problems. In this paper, we recast the maximum vertex weight clique problem (MVWCP) into this model which is then solved by a probabilistic tabu search algorithm designed for the BQP. Experimental results on 80 challenging DIMACS-W and 40 BHOSLIB-W benchmark instances demonstrate that this general approach is viable for solving the MVWCP problem.

Keywords

Maximum vertex weight clique Binary quadratic programming Probabilistic tabu search 

References

  1. Alidaee B, Glover F, Kochenberger GA, Wang H (2007) Solving the maximum edge weight clique problem via unconstrained quadratic programming. Eur J Oper Res 181:592–597CrossRefMATHGoogle Scholar
  2. Alidaee B, Kochenberger GA, Lewis K, Lewis M, Wang H (2008) A new approach for modeling and solving set packing problem. Eur J Oper Res 86(2):504–512MathSciNetCrossRefMATHGoogle Scholar
  3. Babel L (1994) A fast algorithm for the maximum weight clique problem. Computing 52(1):31–38MathSciNetCrossRefMATHGoogle Scholar
  4. Ballard D, Brown C (1983) Computer vision. Prentice-Hall, Englewood CliffsGoogle Scholar
  5. Benlic U, Hao JK (2013) Breakout local search for maximum clique problems. Comput Oper Res 40(1):192–206MathSciNetCrossRefMATHGoogle Scholar
  6. Bomze IM, Pelillo M, Stix V (2000) Approximating the maximum weight clique using replicator dynamics. IEEE Trans Neural Netw 11:1228–1241CrossRefGoogle Scholar
  7. Busygin S (2006) A new trust region technique for the maximum weight clique problem. Discret Appl Math 154:2080–2096MathSciNetCrossRefMATHGoogle Scholar
  8. Carraghan R, Pardalos PM (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9(6):375–382CrossRefMATHGoogle Scholar
  9. Dorigo M (1997) Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans Evolut Comput 1(1):53–66CrossRefGoogle Scholar
  10. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-Completeness. Freeman, San FranciscoMATHGoogle Scholar
  11. Glover F (1989) Tabu search—Part I. ORSA J Comput 1(3):190–206CrossRefMATHGoogle Scholar
  12. Glover F, Hao JK (2010) Efficient evaluation for solving 0–1 unconstrained quadratic optimization problems. Int J Metaheuristics 1(1):3–10MathSciNetCrossRefMATHGoogle Scholar
  13. Glover F, Hao JK (2010) Fast 2-flip move evaluations for binary unconstrained quadratic optimization problems. Int J Metaheuristics 1(2):100–107MathSciNetCrossRefMATHGoogle Scholar
  14. Glover F, Laguna M (1997) Tabu search. Kluwer Academic Publishers, NorwellCrossRefMATHGoogle Scholar
  15. Hansen P, Mladenović N (2001) Variable neighborhood search: principles and applications. Eur J Oper Res 130(3):449–467MathSciNetCrossRefMATHGoogle Scholar
  16. He K, Huang W (2010) A quasi-human algorithm for solving the three-dimensional rectangular packing problem. Sci China Inf Sci 53(12):2389–2398MathSciNetCrossRefMATHGoogle Scholar
  17. Horst R, Pardalos PM, Thoai NV (1995) Introduction to global optimization, nonconvex optimization and its applications, vol 3. Kluwer Academic Publishers, NorwellMATHGoogle Scholar
  18. Kochenberger GA, Glover F, Alidaee B, Rego C (2004) A unified modeling and solution framework for combinatorial optimization problems. OR Spectr 26:237–250CrossRefMATHGoogle Scholar
  19. Kochenberger G, Alidaee B, Glover F, Wang HB (2007) An effective modeling and solution approach for the generalized independent set problem. Optim Lett 1:111–117MathSciNetCrossRefMATHGoogle Scholar
  20. Kochenberger G, Hao JK, Lü Z, Wang H, Glover F (2013) Solving large scale max cut problems via tabu search. J Heuristics 19(4):565–571CrossRefGoogle Scholar
  21. Kochenberger G, Hao JK, Glover F, Lewis M, Lü Z, Wang H, Wang Y (2014) The unconstrained binary quadratic programming problem: a survey. J Comb Optim 28(1):58–81MathSciNetCrossRefMATHGoogle Scholar
  22. Konc J, Janĕzic̆ D (2007) An improved branch and bound algorithm for the maximum clique problem. MATCH Commun Math Comput Chem 58:569–590MathSciNetMATHGoogle Scholar
  23. Li C, Quan Z (2010) An efficient branch-and-bound algorithm based on MAXSAT for the maximum clique problem. In: Proceedings of the 24th AAAI conference on artificial intelligence, pp 128–133Google Scholar
  24. Lewis M, Kochenberger G, Alidaee B (2008) A new modeling and solution approach for the set-partitioning problem. Comput Oper Res 2008:807–813MathSciNetCrossRefMATHGoogle Scholar
  25. Macreesh C, Prosser P (2013) Multi-threading a state-of-the-art maximum clique algorithm. Algorithms 6(4):618–635MathSciNetCrossRefGoogle Scholar
  26. Manninno C, Stefanutti E (1999) An augmentation algorithm for the maximum weighted stable set problem. Comput Optim Appl 14:367–381MathSciNetCrossRefMATHGoogle Scholar
  27. Östergård PRJ (2001) A new algorithm for the maximum weight clique problem. Nordic J Comput 8(4):424–436MathSciNetGoogle Scholar
  28. Östergård PRJ (2002) A fast algorithm for the maximum clique problem. Discret Appl Math 120(1):197–207MathSciNetCrossRefGoogle Scholar
  29. Pajouh FM, Balasumdaram B, Prokopyev O (2013) On characterization of maximal independent sets via quadratic optimization. J Heuristics 19(4):629–644CrossRefGoogle Scholar
  30. Pardalos PM, Rodgers GP (1992) A branch and bound algorithm for the maximum clique problem. Comput Oper Res 19(5):363–375CrossRefMATHGoogle Scholar
  31. Pullan W (2008) Approximating the maximum vertex/edge weighted clique using local search. J Heuristics 14:117–134CrossRefMATHGoogle Scholar
  32. Rebennack S, Oswald M, Theis D, Seitz H, Reinelt G, Pardalos PM (2011) A branch and cut solver for the maximum stable set problem. J Comb Optim 21(4):434–457MathSciNetCrossRefMATHGoogle Scholar
  33. Rebennack S, Reinelt G, Pardalos PM (2012) A tutorial on branch and cut algorithms for the maximum stable set problem. Int Trans Oper Res 19(1–2):161–199MathSciNetCrossRefMATHGoogle Scholar
  34. Segundo PS, Rodríguez-Losada D, Jiménez A (2011) An exact bitparallel algorithm for the maximum clique problem. Comput Oper Res 38(2):571–581MathSciNetCrossRefMATHGoogle Scholar
  35. Sengor NS, Cakir Y, Guzelis C, Pekergin F, Morgul O (1999) An analysis of maximum clique formulations and saturated linear dynamical network. ARI 51:268–276CrossRefGoogle Scholar
  36. Tomita E, Kameda T (2007) An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J Glob Optim 37(1):95–111MathSciNetCrossRefMATHGoogle Scholar
  37. Wang Y, Lü Z, Glover F, Hao JK (2013) Probabilistic GRASP-tabu search algorithms for the UBQP problem. Comput Oper Res 40(12):3100–3107MathSciNetCrossRefGoogle Scholar
  38. Warren JS, Hicks IV (2006) Combinatorial branch-and-bound for the maximum weight independent set problem. Technical Report, Texas A&M UniversityGoogle Scholar
  39. Wu Q, Hao JK (2015) A review on algorithms for maximum clique problems. Eur J Oper Res 242:693–709MathSciNetCrossRefGoogle Scholar
  40. Wu Q, Hao JK, Glover F (2012) Multi-neighborhood tabu search for the maximum weight clique problem. Ann Oper Res 196(1):611–634MathSciNetCrossRefMATHGoogle Scholar
  41. Wu Y, Huang W, Lau S, Wong CK, Young GH (2002) An effective quasi-human based heuristic for solving the rectangle packing problem. Eur J Oper Res 141(2):341–358MathSciNetCrossRefMATHGoogle Scholar
  42. Xu JF, Chiu SY, Glover F (1996) Probabilistic tabu search for telecommunications network design. Comb Optim Theory Pract 1(1):69–94Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Yang Wang
    • 1
  • Jin-Kao Hao
    • 2
    • 3
  • Fred Glover
    • 4
  • Zhipeng Lü
    • 5
  • Qinghua Wu
    • 6
  1. 1.School of ManagementNorthwestern Polytechnical UniversityXi’anChina
  2. 2.LERIAUniversité d’AngersAngersFrance
  3. 3.Institut Universitaire de FranceParisFrance
  4. 4.OptTek Systems, IncBoulderUSA
  5. 5.School of Computer Science and TechnologyHuazhong University of Science and TechnologyWuhanChina
  6. 6.School of ManagementHuazhong University of Science and TechnologyWuhanChina

Personalised recommendations