Optimization Letters

, Volume 9, Issue 5, pp 839–843 | Cite as

The clique problem for graphs with a few eigenvalues of the same sign

  • D. S. MalyshevEmail author
  • P. M. Pardalos
Original Paper


The quadratic programming problem is known to be NP-hard for Hessian matrices with only one negative eigenvalue, but it is tractable for convex instances. These facts yield to consider the number of negative eigenvalues as a complexity measure of quadratic programs. We prove here that the clique problem is tractable for two variants of its Motzkin-Strauss quadratic formulation with a fixed number of negative eigenvalues (with multiplicities).


Quadratic programming Computational complexity Clique problem 



Research is partially supported by LATNA laboratory, National Research University Higher School of Economics, RF government grant, ag. 11.G34.31.00357.


  1. 1.
    Abello, J., Butenko, S., Pardalos, P., Resende, M.: Finding independent sets in a graph using continuous multivariable polynomial formulations. J. Glob. Optim. 21, 111–137 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Balas, E., Yu, C.: On graphs with polynomially solvable maximum weight clique problem. Networks 19, 247–253 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cvetković, D., Doob, M., Sachs, H.: Spectra of Graphs: Theory and Applications. V.E.B. Deutscher Verlag der Wissenschaften, Berlin (1979)Google Scholar
  4. 4.
    Hager, W., Pardalos, P., Roussos, I., Sahinoglou, D.: Active constraints, indefinite quadratic test problems, and complexity. J. Optim. Theory Appl. 68, 499–511 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Kozlov, M., Tarasov, S., Khachiyan, L.: The polynomial solvability of convex quadratic programming. USSR Comput. Math. Math. Phys. 20, 223–228 (1979)CrossRefGoogle Scholar
  6. 6.
    Marcus, M., Minc, H.: Survey of matrix theory and matrix inequalities. Allyn and Bacon, Boston (1964)zbMATHGoogle Scholar
  7. 7.
    Motzkin, T., Strauss, E.: Maxima for graphs and a new proof of a theorem of Turán. Can. J. Math. 17, 533–540 (1965)CrossRefzbMATHGoogle Scholar
  8. 8.
    Pardalos, P., Vavasis, S.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 1, 15–22 (1991)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsUlitsaRussia
  2. 2.University of FloridaGainesvilleUSA

Personalised recommendations