The Maximum Ratio Clique Problem: A Continuous Optimization Approach and Some New Results

  • Mahdi MoeiniEmail author
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 359)


In this paper, we are interested in studying the maximum ratio clique problem (MRCP) that is a variant of the classical maximum weight clique problem. For a given graph, we suppose that each vertex of the graph is weighted by a pair of rational numbers. The objective of MRCP consists in finding a maximal clique with the largest ratio between two sets of weights that are assigned to its vertices. It has been proven that the decision version of this problem is NP-complete and it is hard to solve MRCP for large instances. Hence, this paper looks for introducing an efficient approach based on Difference of Convex functions (DC) programming and DC Algorithm (DCA) for solving MRCP. Then, we verify the performance of the proposed method. For this purpose, we compare the solutions of DCA with the previously published results. As a second objective of this paper, we identify some valid inequalities and evaluate empirically their influence in solving MRCP. According to the numerical experiments, DCA provides promising and competitive results. Furthermore, the introduction of the valid inequalities improves the computational time of the classical approaches.


Maximum Ratio Clique Problem Fractional Programming DC Programming DCA 


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  1. 1.
    Boginski, V., Butenko, S., Pardalos, P.: Mining market data: a network approach. Comput. Oper. Res. 33, 3171–3184 (2006)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ibaraki, T.: Parametric approaches to fractional programs. Math. Prog. 26, 345–362 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Isbell, J.R., Marlow, W.H.: Attrition games. Naval Res. Logist. Q 3(1-2), 71–94 (1956)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Kröller, A., Moeini, M., Schmidt, C.: A Novel Efficient Approach for Solving the Art Gallery Problem. In: Ghosh, S.K., Tokuyama, T. (eds.) WALCOM 2013. LNCS, vol. 7748, pp. 5–16. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Lawler, E.L.: Combinatorial optimization: networks and matroids. Holt, Rinehart and Winston, New York (1976)zbMATHGoogle Scholar
  6. 6.
    Le Thi, H.A.: Contribution à l’optimisation non convexe et l’optimisation globale: Théorie, Algorithmes et Applications. Habilitation à Diriger des Recherches, Université de Rouen (1997)Google Scholar
  7. 7.
    Le Thi, H.A., Pham Dinh, T.: A continuous approach for globally solving linearly constrained quadratic zero-one programming problems. Optimization 50(1-2), 93–120 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Le Thi, H.A., Pham Dinh, T.: The DC (difference of convex functions) Programming and DCA revisited with DC models of real world non convex optimization problems. Annals of Operations Research 133, 23–46 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Le Thi, H.A., Pham Dinh, T., Huynh, V.N.: Exact Penalty Techniques in DC Programming. Research Report, LMI, INSA-Rouen, France (2005)Google Scholar
  10. 10.
    Le Thi, H.A., Moeini, M.: Portfolio selection under buy-in threshold constraints using DC programming and DCA. In: International Conference on Service Systems and Service Management (IEEE/SSSM 2006), pp. 296–300 (2006)Google Scholar
  11. 11.
    Le Thi, H.A., Moeini, M., Pham Dinh, T.: Portfolio Selection under Downside Risk Measures and Cardinality Constraints based on DC Programming and DCA. Computational Management Science 6(4), 477–501 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Le Thi, H.A., Moeini, M., Pham Dinh, T.: DC Programming Approach for Portfolio Optimization under Step Increasing Transaction Costs. Optimization 58(3), 267–289 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Le Thi, H.A., Moeini, M., Pham Dinh, T., Judice, J.: A DC Programming Approach for Solving the Symmetric Eigenvalue Complementarity Problem. Computational Optimization and Applications 51, 1097–1117 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Le Thi, H.A., Moeini, M.: Long-Short Portfolio Optimization under Cardinality Constraints by Difference of Convex Functions Algorithm. Journal of Optimization Theory and Applications 161(1), 199–224 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Liu, Y., Shen, X., Doss, H.: Multicategory ψ-Learning and Support Vector Machine: Computational Tools. Journal of Computational and Graphical Statistics 14, 219–236 (2005)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Luce, R., Perry, A.: A method of matrix analysis of group structure. Psychometrika 14, 95–116 (1949)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Nalan, G., Le Thi, H.A., Moeini, M.: Robust investment strategies with discrete asset choice constraints using DC programming. Optimization 59(1), 45–62 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Pham Dinh, T., Le Thi, H.A.: Convex analysis approach to d.c. programming: Theory, Algorithms and Applications. Acta Mathematica Vietnamica, dedicated to Professor Hoang Tuy on the occasion of his 70th birthday 22(1), 289–355 (1997)zbMATHGoogle Scholar
  19. 19.
    Pham Dinh, T., Le Thi, H.A.: DC optimization algorithms for solving the trust region subproblem. SIAM J. Optimization 8, 476–505 (1998)CrossRefzbMATHGoogle Scholar
  20. 20.
    Prokopyev, O.A., Huang, H., Pardalos, P.M.: On complexity of unconstrained hyperbolic 0-1 programming problems. Oper. Res. Lett. 3(3), 312–318 (2005)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Sethuraman, S., Butenko, S.: The Maximum Ratio Clique Problem. Comp. Man. Sci. 12(1), 197–218 (2015)MathSciNetGoogle Scholar
  22. 22.
    Wu, T.: A note on a global approach for general 0–1 fractional programming. Eur. J. Oper. Res. 101(1), 220–223 (1997)CrossRefzbMATHGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Chair of Business Information Systems and Operations Research (BISOR)Technical University of KaiserslauternKaiserslauternGermany

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